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On optimality conditions for optimal control problem in coefficients for Δp-Laplacian

Olha P Kupenko12 and Rosanna Manzo3*

Author Affiliations

1 Department of System Analysis and Control, National Mining University, Karl Marx av., 19, Dnipropetrovsk, 49005, Ukraine

2 Institute of Applied and System Analysis, National Technical University of Ukraine ‘Kiev Polytechnical Institute’, Peremogy av., 37, build. 35, Kiev, 03056, Ukraine

3 Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Italy

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Boundary Value Problems 2014, 2014:72  doi:10.1186/1687-2770-2014-72

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/72


Received:9 January 2014
Accepted:11 March 2014
Published:28 March 2014

© 2014 Kupenko and Manzo; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>-coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincaré Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair.

MSC: 35J70, 49J20, 49J45, 93C73.

Keywords:
nonlinear Dirichlet problem; control in coefficients; optimality conditions

1 Introduction

The aim of this paper is to derive a first-order optimality system for a nonlinear Dirichlet optimal control problem where the control is taken as an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M4">View MathML</a>-coefficient in a nonlinear state equation. The optimal control problem we consider in this paper is to minimize the discrepancy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M5">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, Ω is an open bounded Lipschitz domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9">View MathML</a> is a given distribution, and y is the solution of a nonlinear Dirichlet problem by choosing an appropriate coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M10">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian. Namely, we consider the following minimization problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M12">View MathML</a>

(1.1)

subject to the constraints

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M13">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M14">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M15">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> is a class of admissible controls and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M18">View MathML</a>.

Optimal control in coefficients for partial differential equations is a classical subject initiated by Lurie [1,2], Lions [3], Zolezzi [4]. Tartar and Murat [5-7] showed examples of the non-existence for such problems (see e.g.[8] also for the historical development). Since the range of OCPs in coefficients is very wide, including as well optimal shape design problems, optimization of certain evolution systems, some problems originating in mechanics and others, this topic has been widely studied by many authors. In particular, it leads to the possibility to optimize material properties what is extremely important for material sciences. The crucial point is to give the right interpretation of the optimal coefficients in the context of applications (see, for instance, [9,10]). Usually this aspect is closely related with the structural assumptions that have to be considered during the optimization process in terms of constraints. One way of doing so is via proper parametrization of the material, respectively, the coefficients, using mixtures, represented by characteristic functions. This has been pursued by Allaire [11] and many other authors in recent years. Another restriction can be realized via regularity of the coefficients and hard constraints. This procedure has been followed first by Casas [12] for a scalar problem, as one of the first papers in that direction, and later by Haslinger et al.[13] in the context of what has come to be known as Free Material Optimization (FMO). However, most of the results and methods rely on linear PDEs, while only very few articles deal with nonlinear problems, see Kogut [14] and Kogut and Leugering [15]. Another point of interest is degeneration in the coefficients which is typically avoided by assuming lower bounds on the coefficients. However, degeneration occurs genuinely in topology optimization, damage and crack problems. In Kogut and Leugering [16-18] and in Kupenko and Manzo [19] this problem has been considered in the context of linear problems (see also [20]). The nonlinear case was considered in [21-24]. In this article, we extend our results to scalar nonlinear problems, where degeneration occurs already with respect to the states.

Another important point, arising after the solvability of the optimization problem had been proved, is the question as regards optimality conditions. The classical approach to deriving such conditions is based on the Lagrange principle. However, in the case when the control is considered in the coefficients of the main part of the state equation, the classical adjoint system often cannot be directly constructed due to the lack of differential properties of the solution to the boundary value problem with respect to control variables. It was the main reason why Serovajskiy has proposed the concept of the so-called quasi-adjoint system [25] and showed that optimality conditions for the linear elliptic control problem in coefficients can be derived, provided the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M19">View MathML</a> possesses the weakened continuity property. However, the verification of this property is not easy matter even for linear systems. In the case of quasi-linear or nonlinear state equations, we are faced with another problem - the Lagrange functional to the indicated problem is not Gâteaux differentiable at the origin. To overcome this difficulty, Casas and Fernández introduced the special family of perturbed optimal control problems and derived the optimality conditions passing to the limit in optimality conditions for approximating control problems. In order to apply this approach to optimal control problem (1.1)-(1.4) it would suffice to assume the following extra conditions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M21">View MathML</a> that look rather restrictive from physical point of view. The second option coming from the approach of Casas and Fernández is the fact that the linear elliptic equation for the adjoint state is not coercive in general and, hence, the adjoint boundary value problem may admit infinitely many solutions. As a result, the attainability of some solutions is rather a questionable matter. That is why in this paper we concentrate not only on deriving of the adjoint system, but also on formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair.

The paper is organized as follows. In Section 2 we give some preliminaries and prescribe the class of admissible controls to problem (1.1)-(1.4). In Section 3 we analyze the solvability properties of optimal control problem (1.1)-(1.4), using the monotonicity of generalized <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian (see for comparison [26,27]). The aim of Section 4 is to give a collection of preliminary results concerning the differentiability properties of the Lagrange functional associated with problem (1.1)-(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M23">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M24">View MathML</a>

and show that it admits the Gâteaux derivative with respect to the so-called non-degenerate directions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25">View MathML</a> at the point y.

In Section 5 we discuss the formal approach in deriving first-order optimality conditions for optimal control problem (1.1)-(1.4). In order to derive an optimality system, we apply the Lagrange principle. It is well known that the proof of this principle is different for different classes of optimal control problem (see, for instance, [10,12,23,28-30]). The complexity of this procedure significantly depends on the form of the extremal problem under consideration. The procedure is rather simple if the controllable system is described by a linear well-posed controllable boundary value problem, but it becomes much more complicated if the controllable system is either ill-posed or nonlinear and singular.

With that in mind, we introduce the notion of a quasi-adjoint state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M26">View MathML</a> to an optimal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> as a solution of the following Dirichlet boundary value problem for degenerate linear elliptic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M28">View MathML</a>

(1.5)

where the degeneration occurs in a natural way with respect to the states.

This concept was proposed for linear problems by Serovajskiy [25], where it was shown that an optimality system for the optimal control problems in coefficients can be recovered in an explicit form if the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M29">View MathML</a> possesses the so-called weakened continuity property. However, it should be stressed that the fulfilment of this property is not proved for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M31">View MathML</a> and, thus, should be considered as some extra hypothesis. Moreover, from a practical point of view, the verification of the weakened continuity property for quasi-adjoint states is not an easy matter, in general. That is why, in order to derive optimality conditions in the framework of more appropriate assumptions, we provide in Section 6 the analysis of the well-posedness of variational problem (1.5) and describe the asymptotic behavior of its solutions as parameter θ tends to zero. However, in contrast to Casas and Fernandez [29], we do not apply a perturbation of the differential operator that removes the singularity at the origin.

In particular, following the well-known Hardy-Poincaré Inequality, we show that the sequence of quasi-adjoint states <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> can be defined in a unique way (as unique solutions for (1.5)) and is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a> provided, for given distributions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, an optimal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M39">View MathML</a> to the nonlinear Dirichlet boundary value problem (1.3)-(1.4) satisfies the properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M40">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M41">View MathML</a>

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M43">View MathML</a>, and some collection of points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M44">View MathML</a>.

Note that the fulfilment of this assumption is feasible if the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M45">View MathML</a> has a non-degenerate spectrum for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M46">View MathML</a> (for the details, we refer to [17]). The main argument given in Sections 6-7 is to look for the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M47">View MathML</a> of the quasi-adjoint problem (1.5) in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M48">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M49">View MathML</a>. As a result, we show that each of the variational problems for the corresponding quasi-adjoint states has a unique solution, and these solutions form a weakly convergent sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M50">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>. This property suffices in order to establish that the optimality system for problem (1.1)-(1.4) remains valid even if the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M45">View MathML</a> has a degenerate spectrum.

2 Notation and preliminaries

Throughout the paper Ω is a bounded open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M55">View MathML</a> of distributions in Ω is the dual of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M56">View MathML</a>. For real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M57">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M58">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M59">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a> is the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M56">View MathML</a> in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M62">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63">View MathML</a> is the space of distributions of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M64">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M65">View MathML</a> (i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63">View MathML</a> is the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>). As a norm in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a> we can take the following one:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M69">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M70">View MathML</a> be the characteristic function of a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M71">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M72">View MathML</a> be its N-dimensional Lebesgue measure.

For any vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M73">View MathML</a>, the divergence is an element of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M74">View MathML</a> defined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M75">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M76">View MathML</a> denotes the duality pairing between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M79">View MathML</a> denotes the scalar product of two vectors in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M7">View MathML</a>.

Functions with bounded variations. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M81">View MathML</a> be a function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M83">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M84">View MathML</a>.

According to the Radon-Nikodym Theorem, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M85">View MathML</a> then the distribution Df is a measure and there exist a vector-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M86">View MathML</a> and a measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M87">View MathML</a>, singular with respect to the N-dimensional Lebesgue measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M88">View MathML</a> restricted to Ω, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M89">View MathML</a>.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M90">View MathML</a> is said to have a bounded variation in Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M85">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92">View MathML</a> we denote the space of all functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a> with bounded variation, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M94">View MathML</a>.

Under the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92">View MathML</a> is a Banach space. For our further analysis, we need the following properties of BV-functions (see [31]).

Proposition 2.2

(i) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M97">View MathML</a>be a sequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M92">View MathML</a>strongly converging to somefin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>and satisfying condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M100">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M101">View MathML</a>

(ii) for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M103">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M105">View MathML</a>

(iii) for every bounded sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M106">View MathML</a>there exist a subsequence, still denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M107">View MathML</a>, and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M108">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M109">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>.

Admissible Controls and Generalizedp-Laplacian. Let α, β, γ, and m be given positive constants such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M112">View MathML</a>. We define the class of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M114">View MathML</a>

(2.2)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> is a nonempty convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a> with empty topological interior.

We say that a nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M117">View MathML</a> is the generalized p-Laplacian if it has a representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M118">View MathML</a>

or via the pairing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M119">View MathML</a>

It is easy to see that for every admissible control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M120">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M121">View MathML</a> turns out to be coercive, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M122">View MathML</a>

and demi-continuous, where by the demi-continuity property we mean the fulfilment of the implication: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M123">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M125">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M63">View MathML</a> (see [26,32]). Moreover, p-Laplacian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M127">View MathML</a> is a strictly monotone operator for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>. Indeed, having applied the trick

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M129">View MathML</a>

it is easy to check the validity of the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M130">View MathML</a>

As a result, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M131">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M132">View MathML</a>

Then by well-known existence results for nonlinear elliptic equations with strictly monotone demi-continuous coercive operators (see [32,33]), one can conclude: for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a>, the nonlinear Dirichlet boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M135">View MathML</a>

(2.3)

admits a unique weak solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>. Let us recall that a function y is the weak solution of (2.3) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M137">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M138">View MathML</a>

(2.5)

3 Setting of the optimal control problem

We consider the following optimal control problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M139">View MathML</a>

(3.1)

subject to the constraints

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M140">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M141">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M143">View MathML</a> are given distributions.

Hereinafter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M144">View MathML</a> denotes the set of all admissible pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M145">View MathML</a> to optimal control problem (3.1)-(3.3).

Remark 3.1 As was mentioned in the previous section, the characteristic feature of optimal control problem (3.1)-(3.3) is the fact that the set of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> is a convex set with an empty topological interior. As we will see later on, this circumstance entails some technical difficulties in the substantiation of optimality conditions for the given problem.

Let τ be the topology on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M147">View MathML</a> which we define as a product of the strong topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a> and the weak topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>. Further we make use of the following results, which play a key role for the solvability of optimal control problem (3.1)-(3.3) (see [26,32] and [15,21] for comparison).

Proposition 3.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151">View MathML</a>, a weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M152">View MathML</a>to the variational problem (3.2)-(3.3) satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M153">View MathML</a>

(3.4)

Proof To prove the proposition it is enough to put in equality (3.2) as a test function the element y and then use the properties of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> on the left-hand side of the relation and the Cauchy-Bunjakowsky Inequality on the right-hand side. Indeed, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M155">View MathML</a>

Now to get the desired estimate we divide each part of the obtain relation into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M156">View MathML</a> and raise each side to the power <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M157">View MathML</a>. □

Proposition 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M164">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M168">View MathML</a>

by Proposition 2.2(i) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M169">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>. Moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163">View MathML</a>, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M172">View MathML</a>

implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162">View MathML</a>.

To end the proof, it is enough to note that strong convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a> implies, up to a subsequence, convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M177">View MathML</a> almost everywhere in Ω. Hence, by the Lebesgue Theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M178">View MathML</a>

that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M164">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>. Since this conclusion is true for any weakly-∗ convergent subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M181">View MathML</a>, it follows that u is the weak-∗ limit for the whole sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182">View MathML</a>. □

Proposition 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a>is a sequentially compact subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163">View MathML</a>, and it is a sequentially weakly-∗ compact subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182">View MathML</a> be any sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M188">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M182">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M190">View MathML</a>. As a result, the statement immediately follows from Propositions 3.2 and 2.2(iii). □

Proposition 3.4For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151">View MathML</a>the set Ξ is sequentially compact, i.e. for each sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M192">View MathML</a>it can be found a subsequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M193">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M194">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M196">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M199">View MathML</a>is a weak solution to the Dirichlet boundary value problem (3.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M200">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M201">View MathML</a> be an arbitrary sequence of admissible pairs. Then Proposition 3.3 and a priori estimate (3.4) lead to the existence of a subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M202">View MathML</a> and a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M203">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M204">View MathML</a>

(3.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M205">View MathML</a>

(3.6)

Taking into account the inequality (see [34])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M206">View MathML</a>

and definition of the class of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a>, we conclude: there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M208">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M209">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M210">View MathML</a>

(3.7)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M212">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M213">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M214">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M215">View MathML</a>

Hence, passing to the limit in (3.7) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M216">View MathML</a>, we arrive at a conclusion (see [26]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M217">View MathML</a>

As a result, we finally have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M218">View MathML</a>

that is, the limit pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M145">View MathML</a> is an admissible to optimal control problem (3.1)-(3.3). The proof is complete. □

Taking into account Propositions 3.1-3.3, in a similar manner to [15,21], it is easy to conclude the following existence result.

Theorem 3.5The optimal control problem (3.1)-(3.3) admits at least one solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M220">View MathML</a>

4 Auxiliary results

The main goal of this paper is to derive the optimality conditions for optimal control problem (3.1)-(3.3). However, we deal with the case when we cannot apply the well-known classical approach (see, for instance, [35,36]), since for a given distribution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M151">View MathML</a> the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M222">View MathML</a> is not Fréchet differentiable on the class of admissible controls, in general, and the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> has an empty topological interior. With that in mind, we apply the so-called differentiation concept on convex sets and introduce the notion of a quasi-adjoint state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M26">View MathML</a> to an optimal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> that was proposed for linear problems by Serovajskiy [25].

To begin with, we discuss the differentiable properties of the Lagrange functional associated with problem (3.1)-(3.3). Since (3.3) can be seen as a constraint, we define the Lagrangian as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M226">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M227">View MathML</a> is a Lagrange multiplier and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M228">View MathML</a>

For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, let us consider the following sets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M231">View MathML</a>

Clearly, we cannot claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M232">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M162">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M234">View MathML</a>, because convergence of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M235">View MathML</a> to ∇y does not imply, in general, the χ-convergence of subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M236">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M237">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M238">View MathML</a> [[15], p.218]. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M239">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M240">View MathML</a> almost everywhere in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M241">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M242">View MathML</a> whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M243">View MathML</a> for all positive θ small enough. Hence, in this case the convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M244">View MathML</a> fails. In view of this, we make use of the following notion (see [23]).

Definition 4.1 We say that an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a> is a regular point for the Lagrangian (4.1) if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M246">View MathML</a> the direction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M247">View MathML</a> is non-degenerate in the following sense:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M248">View MathML</a>

(4.2)

We have the following result.

Proposition 4.2Assume thatyis an element of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>such that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M250">View MathML</a>

(4.3)

has zero Lebesgue measure. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a>is a regular point of the Lagrangian (4.1) in the sense of Definition 4.1.

Proof Let h be a given element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M253">View MathML</a>, then, by definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M254">View MathML</a>. Thus, there is a value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M255">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M256">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M257">View MathML</a>. It is worth to note that this pointwise inequality makes a sense if only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M258">View MathML</a> is a Lebesgue point of both ∇y and ∇h. However, the Lebesgue Differentiation Theorem states that, given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M259">View MathML</a>, almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M258">View MathML</a> is a Lebesgue point. Hence, almost all Lebesgue points of ∇y are the Lebesgue points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M261">View MathML</a> for θ small enough, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M262">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M263">View MathML</a>. Since the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M264">View MathML</a> has zero Lebesgue measure, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M266">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a> small enough. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M268">View MathML</a> almost everywhere in Ω and hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M268">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>. □

Remark 4.1 It is worth noting that due to the results of Manfredi (see [37]), the assumptions of Proposition 4.2 appears natural and it is not a restrictive supposition in practice. Indeed, following [37], we can ensure that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M271">View MathML</a> for non-constant solutions of the p-Laplace equation (a p-harmonic function) has zero Lebesgue measure. Moreover, it is also easy to observe that if y and v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> are two regular points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M273">View MathML</a>, then there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M274">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M275">View MathML</a>) such that each point of the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M276">View MathML</a> is also regular for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M277">View MathML</a>.

We are now ready to study the differentiability properties of the Lagrangian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M277">View MathML</a>. We begin with the following result.

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>be a given element, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a>be a regular point of the Lagrangian (4.1). Then the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M281">View MathML</a>

is Gâteaux differentiable atyand its Gâteaux derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M282">View MathML</a>

exists and takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M283">View MathML</a>

(4.4)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a> be a regular point for the Lagrangian (4.1) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M25">View MathML</a> be an arbitrary distribution. Following the definition of the Gâteaux derivative, we have to deduce the following equality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M286">View MathML</a>

With that in mind, let us consider the vector-valued function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M287">View MathML</a>

for which the Taylor expansion with the remainder term in the Lagrange form leads to the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M288">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M289">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M290">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M291">View MathML</a> be an arbitrary value. Let us consider the following decomposition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M292">View MathML</a>

where the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M264">View MathML</a> is defined by (4.3), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M295">View MathML</a> are measurable subsets of Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M296">View MathML</a>

Following [[34], p.598] (see also [38]), we have: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M297">View MathML</a> there exists a positive value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M298">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M299">View MathML</a>

(4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M300">View MathML</a>

(4.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M302">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M303">View MathML</a> small enough. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M304">View MathML</a> by the initial assumptions, it follows from (4.5)-(4.6) that the vector-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M305">View MathML</a> is Gâteaux differentiable. Hence, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M306">View MathML</a> is Gâteaux differentiable for any regular point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a> and for any admissible control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M308">View MathML</a>, and its Gâteaux derivative takes the form (4.4). □

Since Gâteaux differentiability of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M309">View MathML</a> implies the existence of Gâteaux derivative for the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M310">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M311">View MathML</a>

we arrive at the following obvious consequence of Lemma 4.3.

Corollary 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M313">View MathML</a>be given elements, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a>be a regular point of the Lagrangian (4.1). Then the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M315">View MathML</a>

is Gâteaux differentiable atyand its Gâteaux derivative,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M316">View MathML</a>

exists and takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M317">View MathML</a>

(4.7)

Remark 4.2 In view of the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M318">View MathML</a>

the last term in (4.7) can be rewritten as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M319">View MathML</a>

Before deriving the optimality conditions, we need the following auxiliary result.

Lemma 4.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M245">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M246">View MathML</a>be given distributions. Assume that each point of the segment<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M323">View MathML</a>is regular for the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M324">View MathML</a>. Then there exists a positive value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M325">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M326">View MathML</a>

(4.8)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M247">View MathML</a>.

Proof For given u, μ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M328">View MathML</a>, y, and v, let us consider the scalar function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M329">View MathML</a>. Since by Corollary 4.4, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M330">View MathML</a> is Gâteaux differentiable at each point of the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M331">View MathML</a>, it follows that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M332">View MathML</a> is differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M333">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M334">View MathML</a>

To conclude the proof, it remains to take into account (4.7) and apply the Mean Value Theorem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M335">View MathML</a>

 □

5 Formalism of the quasi-adjoint technique

We begin with the following assumption:

(H1) The distribution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a> is such that, for each admissible control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a>, the corresponding weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M338">View MathML</a> of the nonlinear Dirichlet boundary value problem (2.3) is a regular point of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M339">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198">View MathML</a> be an optimal pair for problem (3.1)-(3.3). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M341">View MathML</a>

(5.1)

Hence, splitting it in two terms, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M342">View MathML</a>

(5.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M345">View MathML</a>.

Since the set of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M346">View MathML</a> has an empty topological interior, we justify the choice of perturbation for an optimal control as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M347">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M348">View MathML</a> is an arbitrary admissible pair, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>. Then due to Hypothesis (H1) and Remark 4.1, we can suppose that each point of the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M350">View MathML</a> is regular for the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M351">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M352">View MathML</a> is the corresponding solution of boundary value problem (3.2)-(3.3). Hence, by Lemma 4.5, there exists a value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353">View MathML</a> such that condition (5.1) can be represented as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M354">View MathML</a>

(5.3)

Using (4.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M355">View MathML</a>

(5.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356">View MathML</a>.

Now we introduce the concept of quasi-adjoint states that was first considered for linear problems by Serovajskiy [25].

Definition 5.1 We say that, for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358">View MathML</a>, a distribution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359">View MathML</a> is the quasi-adjoint state to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359">View MathML</a> satisfies the following integral identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M362">View MathML</a>

(5.5)

or in the operator form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M363">View MathML</a>

(5.6)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M364">View MathML</a> is the identity matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365">View MathML</a> is the solution of problem (3.2)-(3.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M367">View MathML</a> is a constant coming from equality (5.3).

Let us assume, for a moment, that quasi-adjoint state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> is a distribution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>. Then we can define the element λ in (5.4) as the quasi-adjoint state, that is, we can set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M371">View MathML</a>. As a result, the increment of Lagrangian (5.4) can be simplified to the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M372">View MathML</a>

(5.7)

Thus, in order to derive the necessary optimality conditions and provide their substantial analysis, it remains to pass to the limit in (5.5)-(5.7) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M373">View MathML</a>, and to show that the sequence of quasi-adjoint states <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M374">View MathML</a> is defined in a unique way through relation (5.5) and it is compact with respect to the weak topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>.

6 The Hardy Inequality and asymptotic behavior of quasi-adjoint states

The main goal of this section is to study the well-posedness of variational problem (5.5) and describe the asymptotic behavior of its solutions as parameter θ tends to zero.

We begin with the following evident consequence of Proposition 3.4.

Lemma 6.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M376">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M159">View MathML</a>strongly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>. Then, for the corresponding solutions of boundary value problem (3.2)-(3.3), we have strong convergence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M379">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>.

Since, by the initial suppositions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>, we immediately arrive at the following consequence of Lemma 6.1.

Corollary 6.2

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M384">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M387">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M388">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>.

Further, we note that

(A1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M390">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M392">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M393">View MathML</a>

(6.1)

and, hence, by the Hölder Inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M394">View MathML</a>

(6.2)

(A2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M396">View MathML</a> then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M397">View MathML</a>

(6.3)

and, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M398">View MathML</a>

(6.4)

Then, in view of Corollary 6.2 and estimates (6.2) and (6.4), we can give the following obvious conclusion.

Corollary 6.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M399">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M401">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>.

Our next intention is to study the variational problem (5.5). With that in mind, we rewrite it in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M404">View MathML</a>

(6.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M405">View MathML</a>

(6.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M406">View MathML</a>

(6.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M407">View MathML</a>

(6.8)

To begin with, we make use of the following observation.

Proposition 6.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198">View MathML</a>be an optimal pair for problem (3.1)-(3.3). Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410">View MathML</a>, we have:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M411">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M412">View MathML</a>denotes the set of all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M413">View MathML</a>symmetric matrices, which are obviously determined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M414">View MathML</a>scalars.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M415">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M416">View MathML</a>.

Proof The first property is obvious. To prove the second one, it enough to take into account the definition of the class of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M16">View MathML</a> and the following chain of estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M418">View MathML</a>

 □

Proposition 6.5

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M419">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M82">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M422">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M423">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M163">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M425">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M426">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427">View MathML</a>;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M428">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M429">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427">View MathML</a>.

Proof Validity of assertions (a)-(b) immediately follows from Corollary 6.3 and Propositions 3.2 and 6.4. The strong convergence property in (c) is a direct consequence of the Lebesgue Convergence Theorem. □

The following results are crucial for our further analysis.

Lemma 6.6Assume that, for given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M433">View MathML</a>. Then each elementψof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>can be represented in a unique way as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M435">View MathML</a>

(6.9)

Proof Let us fix an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M436">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M437">View MathML</a>, by the Hölder Inequality with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M438">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M439">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M440">View MathML</a>

where the constant C comes from the Poincaré Inequality. Using the evident equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M441">View MathML</a>

and applying the Hölder Inequality with exponents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M442">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M443">View MathML</a> for the first term and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M438">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M439">View MathML</a> for the second one, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M446">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M447">View MathML</a>. Since the element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M437">View MathML</a> inherits the trace properties along Ω from its parent element ψ, we finally obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M449">View MathML</a>. The proof is complete. □

As an obvious consequence of this result and continuity of the embedding of Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M450">View MathML</a>, we can give the following conclusion.

Corollary 6.7If, for given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M453">View MathML</a>, then there exists a dense subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M454">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M456">View MathML</a>

(6.10)

Taking this fact into account, it is plausible to introduce the following linear mapping:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M457">View MathML</a>

(6.11)

Since domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M454">View MathML</a> of is dense in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>, it follows that for , as for a densely defined operator, there exists an adjoint operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M462">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M463">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M464">View MathML</a>

Notice that, in general, the adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M465">View MathML</a> is not densely defined.

Let us consider the following linear operator:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M466">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M467">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M468">View MathML</a> are defined by (6.6)-(6.7). By Proposition 6.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M469">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M470">View MathML</a> and this operator is obviously monotone,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M471">View MathML</a>

and demi-continuous. However, because of the multiplier <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M472">View MathML</a>, this operator can lose the coercivity property.

Let λ be a positive constant such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M473">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M474">View MathML</a> be a given collection of points. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M128">View MathML</a> be a given control. We define a nonempty subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M476">View MathML</a> as follows: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M477">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M478">View MathML</a>

(6.12)

for some positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M480">View MathML</a>

(6.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M481">View MathML</a>

(6.14)

We are now in a position to give an important property of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M470">View MathML</a>.

Lemma 6.8Assume that, for given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353">View MathML</a>, the distribution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M486">View MathML</a>is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M487">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M488">View MathML</a>

(6.15)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M489">View MathML</a>

(6.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M490">View MathML</a>

(6.17)

and the linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M491">View MathML</a>defines an isomorphism from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>into its dual<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M493">View MathML</a>.

Proof Let v and z be arbitrary elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M494">View MathML</a>. Then by Corollary 6.7, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M495">View MathML</a>. Further, following the definition of the operator , we can provide the following chain of transformations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M497">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M498">View MathML</a>

To conclude the proof it remains to show that operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M499">View MathML</a> defines an isomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a> into its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M501">View MathML</a>. With that in mind, we make use of the following version of the Hardy-Poincaré Inequality: for a given internal point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M502">View MathML</a> there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M42">View MathML</a> such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M504">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M505">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M506">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M8">View MathML</a>.

According to this result and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M508">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M509">View MathML</a>

(6.18)

and, therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M510">View MathML</a>

(6.19)

Thus, in view of (6.18)-(6.19),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M511">View MathML</a>

is equivalent to the standard norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>, and therefore, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M491">View MathML</a> given by (6.16) defines an isomorphism from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a> into its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M493">View MathML</a>. □

The next step of our analysis is to show that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M516">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353">View MathML</a>, the quasi-adjoint state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> can be defined as a unique solution to the Dirichlet boundary value problem (6.5). With that in mind, we make use of the following hypothesis.

(H2) For a given distribution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M522">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, the weak solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M524">View MathML</a> of the nonlinear Dirichlet boundary value problem (2.3) satisfy the property: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M525">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M526">View MathML</a> for every pair of admissible controls <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M527">View MathML</a>.

Lemma 6.9Assume that the Hypothesis (H2) is valid. Then the Dirichlet boundary value problem (6.5) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M528">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M198">View MathML</a> be an optimal pair to problem (3.1)-(3.3). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365">View MathML</a> be the solution of problem (3.2)-(3.3) for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M356">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M537">View MathML</a> takes an arbitrary value in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M333">View MathML</a>. Since, by the initial assumptions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M539">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M540">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M541">View MathML</a>. Then Lemma 6.8 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M542">View MathML</a>

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M543">View MathML</a>. By the Hardy-Poincaré Inequality (see (6.19)), the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M544">View MathML</a>

can be considered as a scalar product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>. Then, in view of the Riesz Representation Theorem, we conclude to the existence of a unique element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M547">View MathML</a>

As a result, we have: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M548">View MathML</a> is a unique solution to the Dirichlet boundary value problem (2.3). Moreover, by Corollary 6.7, we finally get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M549">View MathML</a>. □

In view of Lemma 6.9, it makes a sense to accept the following hypothesis.

(H3) For given distributions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M522">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>, the weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M524">View MathML</a> to the nonlinear Dirichlet boundary value problem (2.3) satisfies the properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M555">View MathML</a>

Lemma 6.10Assume that Hypotheses (H2)-(H3) are valid. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M556">View MathML</a>independent ofθsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M557">View MathML</a>, i.e. the sequence of quasi-adjoint states<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M558">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a>is relatively compact with respect to the weak convergence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>.

Proof Due to Hypothesis (H2) and Lemma 6.9, the sequence of quasi-adjoint states <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M561">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> can be defined in a unique way <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M563">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M565">View MathML</a> is the solution of the following Dirichlet problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M566">View MathML</a>

(6.20)

Therefore, in order to prove this lemma, it is enough to show that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M567">View MathML</a> is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M455">View MathML</a>. To this end, we note that the integral identity (6.20) leads to the corresponding energy equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M569">View MathML</a>

(6.21)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M570">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M571">View MathML</a>

it follows that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M572">View MathML</a> independent of θ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M573">View MathML</a>

As a result, in view of (6.19), the energy equality (6.21) immediately leads to the estimate.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M574">View MathML</a>

This concludes the proof. □

7 Optimality conditions

We are now in a position to derive the first-order optimality conditions for optimal control problem (3.1)-(3.3).

Theorem 7.1Let us suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M576">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M577">View MathML</a>are given with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M6">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M579">View MathML</a>be an optimal pair to problem (3.1)-(3.3). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M359">View MathML</a>be a quasi-adjoint state to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a>defined for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M353">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>by (5.5). Then Hypotheses (H1)-(H3) imply the existence of an element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M584">View MathML</a>such that (within a subsequence) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M585">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M427">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M588">View MathML</a>

(7.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M589">View MathML</a>

(7.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M590">View MathML</a>

(7.3)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M591">View MathML</a> be an admissible pair. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M365">View MathML</a> be the solution of problem (3.2)-(3.3) for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M358">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M230">View MathML</a>. Then, as was shown at the end of Section 5, there exists a value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M595">View MathML</a> such that the increment of Lagrangian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M596">View MathML</a> can be simplified to the form (5.7), provided the element λ has been defined as the quasi-adjoint state, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M371">View MathML</a>. By Lemmas 6.9-6.10, the sequence of quasi-adjoint states <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M27">View MathML</a> can be defined in a unique way and is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a>. Hence, there exists an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M436">View MathML</a> such that, up to a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M32">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M603">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M34">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>. It remains to pass to the limit in (5.5), (5.7) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M373">View MathML</a> and to show that in the limit we will arrive at the relations (7.3) and (7.1), respectively. To this end, we note that

(A1) by the initial suppositions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>;

(A2) by Corollary 6.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M384">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>, and, hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M613">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M383">View MathML</a>;

(A3) up to a subsequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M616">View MathML</a>

by the strong convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M381">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a> and convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M387">View MathML</a> almost everywhere in Ω;

(A4) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M620">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M621">View MathML</a>

(7.4)

(A5) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M622">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M623">View MathML</a>

(7.5)

Then the limit passage in (5.7) immediately leads to (7.1). Therefore, in order to end the proof, it remains to establish the validity of relation (7.3). With that in mind, we rewrite (5.5) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M624">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M625">View MathML</a>

and φ is an arbitrary element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M627">View MathML</a>

let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M628">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M629">View MathML</a>), and, hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M630">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M631">View MathML</a>.

Using the Hölder Inequality and estimate (2) in Proposition 6.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M632">View MathML</a>

(7.6)

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M620">View MathML</a>, then, by (A4) and Hölder’s Inequality with Hölder conjugates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M634">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M635">View MathML</a>, we can estimate (7.6) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M636">View MathML</a>

(7.7)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M637">View MathML</a> by Lemma 6.10, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M638">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> (see the condition (A2)), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M640">View MathML</a>

Thus, in view of estimate (7.7), we conclude: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M641">View MathML</a>.

As for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M622">View MathML</a>, the inequality (7.6) and condition (A5) lead to the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M643">View MathML</a>

Further it remains to repeat the trick like in (7.7). As a result, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M644">View MathML</a>

Therefore, having applied the arguments given before, we can conclude: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M645">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M646">View MathML</a>.

As for the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M647">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M648">View MathML</a>

(7.8)

To clarify the asymptotic behavior of the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M649">View MathML</a> as θ tends to zero, we note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M650">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M651">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M652">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M653">View MathML</a>, it follows that the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M654">View MathML</a> holds true with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M18">View MathML</a>. Hence, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M656">View MathML</a> is ensured by the weak convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M657">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M658">View MathML</a>.

Thus, summing up the results given above, we finally obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M659">View MathML</a>

(7.9)

As for the term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M660">View MathML</a>

we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M661">View MathML</a>

Hence, strong convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M662">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M124">View MathML</a> implies strong convergence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M664">View MathML</a>

(7.10)

As a result, we finally get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M665">View MathML</a>

(7.11)

Thus, combining relations (7.9)-(7.11), it is easy to see that the limit passage in (5.5) leads to the variational statement of the Dirichlet boundary value problem (7.3). The proof is complete. □

8 Conclusions

In this paper the optimal control problem (3.1)-(3.3) for a nonlinear monotone elliptic equation with homogeneous Dirichlet conditions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>-control in coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian has been studied. Having defined the class of admissible control in form (2.2), we have proved solvability of the considered problem. After that, using Lagrange principle, the concept of quasi-adjoint system and the well-known Hardy-Poincaré Inequality, we have derived the corresponding optimality system and formulated sufficient conditions under which the degenerate adjoint boundary value problem admits a unique solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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