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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 L ( Ω ) -coefficient of Δ p -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 L -coefficient in a nonlinear state equation. The optimal control problem we consider in this paper is to minimize the discrepancy y y d W 0 1 , p ( Ω ) p , where p 2 , Ω is an open bounded Lipschitz domain in R N with N 2 , y d W 0 1 , p ( Ω ) is a given distribution, and y is the solution of a nonlinear Dirichlet problem by choosing an appropriate coefficient u L ( Ω ) of Δ p -Laplacian. Namely, we consider the following minimization problem:

Minimize  { I ( u , y ) = Ω | y ( x ) y d ( x ) | p d x } (1.1)

subject to the constraints

u A ad L ( Ω ) B V ( Ω ) , y W 0 1 , p ( Ω ) , (1.2)

div ( u | y | p 2 y ) = f in  Ω , (1.3)

y = 0 on  Ω , (1.4)

where A ad is a class of admissible controls and f W 1 , q ( Ω ) , q = p / ( p 1 ) .

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 u ψ ε ( u ) 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: p > N / 2 and A ad C 1 ( Ω ) 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 Δ p -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)

Λ ( u , y , λ ) = I ( u , y ) + a u ( y , λ ) f , λ W 0 1 , p ( Ω ) ,

where

a u ( y , λ ) = Ω u ( x ) | y | p 2 ( y , λ ) R N d x

and show that it admits the Gâteaux derivative with respect to the so-called non-degenerate directions h W 0 1 , p ( Ω ) 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 ψ ε to an optimal solution y 0 W 0 1 , p ( Ω ) as a solution of the following Dirichlet boundary value problem for degenerate linear elliptic equation:

div ( u θ | y θ | p 2 [ I + ( p 2 ) y θ | y θ | y θ | y θ | ] ψ θ ) = p div ( | y θ y d | p 2 ( y θ y d ) ) in  Ω , ψ θ W 0 1 , p ( Ω ) , (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 ad u ψ ε ( u ) 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 Δ p -Laplacian with p > 2 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 { ψ ε θ , θ } θ 0 to y 0 W 0 1 , p ( Ω ) can be defined in a unique way (as unique solutions for (1.5)) and is bounded in W 0 1 , p ( Ω ) provided, for given distributions f W 1 , q ( Ω ) and y d W 0 1 , p ( Ω ) with q = p p 1 and p 2 , an optimal solution y 0 = y ( u 0 ) to the nonlinear Dirichlet boundary value problem (1.3)-(1.4) satisfies the properties

ln | y 0 | L 2 ( Ω ; R N ) , ( | y 0 | ) L p ( Ω ; R N ) and | y d | | y 0 | L ( Ω ) , C ˆ ( Ω ) V ( x ) 2 λ L i = 1 L 1 | x x i | 2 a.e. in  Ω ,

where

V ( x ) = ( 2 p ) div ( A ( u 0 , y 0 ) ln | y 0 | ) ( p 2 ) 2 2 ( ln | y 0 | , A ( u 0 , y 0 ) ln | y 0 | ) R N , A ( u 0 , y 0 ) = u 0 [ I + ( p 2 ) y 0 | y 0 | y 0 | y 0 | ] ,

for some positive constants C ˆ ( Ω ) > 0 , λ < λ : = ( N 2 ) 2 / 4 , and some collection of points { x 1 , x 2 , , x L } Ω .

Note that the fulfilment of this assumption is feasible if the matrix | y | p 2 A ( u , y ) has a non-degenerate spectrum for each U U ad (for the details, we refer to [17]). The main argument given in Sections 6-7 is to look for the solutions ψ θ W 0 1 , p ( Ω ) of the quasi-adjoint problem (1.5) in the form ψ θ = | y θ | ( 2 p ) / 2 z θ , where z θ H 0 1 ( Ω ) . 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 { ψ θ } θ 0 in W 0 1 , p ( Ω ) . This property suffices in order to establish that the optimality system for problem (1.1)-(1.4) remains valid even if the matrix | y | p 2 A ( u , y ) has a degenerate spectrum.

2 Notation and preliminaries

Throughout the paper Ω is a bounded open subset of R N , N 2 . The space D ( Ω ) of distributions in Ω is the dual of the space C 0 ( Ω ) . For real numbers 2 p < + , and 1 < q < + such that 1 / p + 1 / q = 1 , the space W 0 1 , p ( Ω ) is the closure of C 0 ( Ω ) in the Sobolev space W 1 , p ( Ω ) , while W 1 , q ( Ω ) is the space of distributions of the form f = f 0 + j D j f j , with f 0 , f 1 , , f n L q ( Ω ) (i.e. W 1 , q ( Ω ) is the dual space of W 0 1 , p ( Ω ) ). As a norm in the space W 0 1 , p ( Ω ) we can take the following one:

y W 0 1 , p ( Ω ) = ( Ω | y | R N p ) 1 / p .

Let χ E be the characteristic function of a set E R N and let | E | be its N-dimensional Lebesgue measure.

For any vector field v L q ( Ω ; R N ) , the divergence is an element of the space W 1 , q ( Ω ) defined by the formula

div v , φ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = Ω ( v , φ ) R N d x , φ W 0 1 , p ( Ω ) , (2.1)

where , W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) denotes the duality pairing between W 1 , q ( Ω ) and W 0 1 , p ( Ω ) , and ( , ) R N denotes the scalar product of two vectors in R N .

Functions with bounded variations. Let f : Ω R be a function of L 1 ( Ω ) . Define

T V ( f ) : = Ω | D f | = sup { Ω f ( , φ ) R N d x : φ C 0 1 ( Ω ; R N ) , | φ ( x ) | 1  for  x Ω } ,

where ( , φ ) R N = i = 1 N φ i x i .

According to the Radon-Nikodym Theorem, if T V ( f ) < + then the distribution Df is a measure and there exist a vector-valued function f L 1 ( Ω ; R N ) and a measure D s f , singular with respect to the N-dimensional Lebesgue measure L N Ω restricted to Ω, such that D f = f L N Ω + D s f .

Definition 2.1 A function f L 1 ( Ω ) is said to have a bounded variation in Ω if T V ( f ) < + . By B V ( Ω ) we denote the space of all functions in L 1 ( Ω ) with bounded variation, i.e. B V ( Ω ) = { f L 1 ( Ω ) : T V ( f ) < + } .

Under the norm f B V ( Ω ) = f L 1 ( Ω ) + T V ( f ) , B V ( Ω ) is a Banach space. For our further analysis, we need the following properties of BV-functions (see [31]).

Proposition 2.2

(i) Let { f k } k = 1 be a sequence in B V ( Ω ) strongly converging to somefin L 1 ( Ω ) and satisfying condition sup k N T V ( f k ) < + . Then

f B V ( Ω ) and T V ( f ) lim inf k T V ( f k ) ;

(ii) for every f B V ( Ω ) L r ( Ω ) , r [ 1 , + ) , there exists a sequence { f k } k = 1 C ( Ω ) such that

lim k Ω | f f k | r d x = 0 and lim k T V ( f k ) = T V ( f ) ;

(iii) for every bounded sequence { f k } k = 1 B V ( Ω ) there exist a subsequence, still denoted by f k , and a function f B V ( Ω ) such that f k f in L 1 ( Ω ) .

Admissible Controls and Generalizedp-Laplacian. Let α, β, γ, and m be given positive constants such that 0 < α β < + and α | Ω | m β | Ω | . We define the class of admissible controls A ad as follows:

A ad = { u B V ( Ω ) L ( Ω ) | T V ( u ) γ , u L 1 ( Ω ) = m , α u ( x ) β  a.e. in  Ω } . (2.2)

It is clear that A ad is a nonempty convex subset of L 1 ( Ω ) with empty topological interior.

We say that a nonlinear operator Δ p : A ad × W 0 1 , p ( Ω ) W 1 , q ( Ω ) is the generalized p-Laplacian if it has a representation

Δ p ( u , y ) = div ( u ( x ) | y | p 2 y ) , where  | y | p 2 : = | y | R N p 2 = ( i = 1 N | y x j | 2 ) p 2 2 ,

or via the pairing

Δ p ( u , y ) , v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = Ω u ( x ) | y | p 2 ( y , v ) R N d x , v W 0 1 , p ( Ω ) .

It is easy to see that for every admissible control u A ad , the operator Δ p ( u , ) turns out to be coercive, i.e.

Δ p ( u , y ) , y W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) α y W 0 1 , p ( Ω ) p ,

and demi-continuous, where by the demi-continuity property we mean the fulfilment of the implication: y k y 0 strongly in W 0 1 , p ( Ω ) implies that Δ p ( u , y k ) Δ p ( u , y 0 ) weakly in W 1 , q ( Ω ) (see [26,32]). Moreover, p-Laplacian Δ p ( u , y ) is a strictly monotone operator for each u A ad . Indeed, having applied the trick

Ω | y | p 1 | v | u ( x ) d x ( Ω ( | y | p 1 u p 1 p ) p p 1 d x ) ( p 1 ) / p ( Ω ( | v | u 1 p ) p d x ) 1 / p = ( Ω | y | p u d x ) ( p 1 ) / p ( Ω | v | p u d x ) 1 / p = : y W 0 1 , p ( Ω ; u d x ) p 1 v W 0 1 , p ( Ω ; u d x ) ,

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

Δ p ( u , y ) Δ p ( u , v ) , y v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = Ω u ( x ) ( | y | p 2 y | v | p 2 v , y v ) R N d x ( by the Cauchy Inequality ) Ω | y | p 2 ( | y | 2 | v | | y | ) u ( x ) d x Ω | v | p 2 ( | v | | y | | v | 2 ) u ( x ) d x ( by the Hölder Inequality ) y W 0 1 , p ( Ω ; u d x ) p + v W 0 1 , p ( Ω ; u d x ) p y W 0 1 , p ( Ω ; u d x ) p 1 v W 0 1 , p ( Ω ; u d x ) v W 0 1 , p ( Ω ; u d x ) p 1 y W 0 1 , p ( Ω ; u d x ) = ( y W 0 1 , p ( Ω ; u d x ) p 1 v W 0 1 , p ( Ω ; u d x ) p 1 ) ( y W 0 1 , p ( Ω ; u d x ) v W 0 1 , p ( Ω ; u d x ) ) 2 2 p | y W 0 1 , p ( Ω ; u d x ) v W 0 1 , p ( Ω ; u d x ) | p , y , v W 0 1 , p ( Ω ) .

As a result, since β 1 y W 0 1 , p ( Ω ; u d x ) p y W 0 1 , p ( Ω ) p α 1 y W 0 1 , p ( Ω ; u d x ) p , it follows that

Δ p ( u , y ) Δ p ( u , v ) , y v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) > 0 for all  y , v W 0 1 , p ( Ω ) , y v .

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 u A ad and f W 1 , q ( Ω ) , the nonlinear Dirichlet boundary value problem

Δ p ( u , y ) = f in  Ω , y W 0 1 , p ( Ω ) , (2.3)

admits a unique weak solution in W 0 1 , p ( Ω ) . Let us recall that a function y is the weak solution of (2.3) if

y W 0 1 , p ( Ω ) , (2.4)

Ω u ( x ) | y | p 2 ( y , v ) R N d x = f , v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) , v W 0 1 , p ( Ω ) . (2.5)

3 Setting of the optimal control problem

We consider the following optimal control problem:

Minimize  { I ( u , y ) = Ω | y ( x ) y d ( x ) | p d x } , (3.1)

subject to the constraints

Ω u ( x ) | y | p 2 ( y , v ) R N d x = f , v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) , v W 0 1 , p ( Ω ) , (3.2)

u A ad L ( Ω ) , y W 0 1 , p ( Ω ) , (3.3)

where f W 1 , q ( Ω ) and y d W 0 1 , p ( Ω ) are given distributions.

Hereinafter, Ξ L ( Ω ) × W 0 1 , p ( Ω ) denotes the set of all admissible pairs ( u , y ) 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 ad 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 L 1 ( Ω ) × W 0 1 , p ( Ω ) which we define as a product of the strong topology of L 1 ( Ω ) and the weak topology of W 0 1 , p ( Ω ) . 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 u A ad and f W 1 , q ( Ω ) , a weak solution y W 0 1 , p ( Ω ) to the variational problem (3.2)-(3.3) satisfies the estimate

y W 0 1 , p ( Ω ) α 1 / ( p 1 ) f W 1 , q ( Ω ) 1 / ( p 1 ) . (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 ad on the left-hand side of the relation and the Cauchy-Bunjakowsky Inequality on the right-hand side. Indeed, we get

α y W 0 1 , p ( Ω ) p Ω u ( x ) | y | p 2 ( y , y ) R N d x = f , y W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) f W 1 , q ( Ω ) y W 0 1 , p ( Ω ) .

Now to get the desired estimate we divide each part of the obtain relation into y W 0 1 , p ( Ω ) and raise each side to the power 1 / ( p 1 ) . □

Proposition 3.2If { u k } k N A ad and u k u in L 1 ( Ω ) , then u k u in L r ( Ω ) for any r [ 1 , + ) and u k u in L ( Ω ) .

Proof Since u k u in L 1 ( Ω ) and

Ω u k d x = m , T V ( u k ) γ ,  and  α u k β  a.e. in  Ω , k N ,

by Proposition 2.2(i) it follows that

T V ( u ) γ , Ω u d x = m ,  and  α u β  a.e. in  Ω .

Hence, u A ad . Moreover, for any r [ 1 , + ) , the estimate

u k u L r ( Ω ) r vrai sup x Ω | u k ( x ) u ( x ) | r 1 u k u L 1 ( Ω ) ( β α ) r 1 u k u L 1 ( Ω )

implies that u k u in L r ( Ω ) .

To end the proof, it is enough to note that strong convergence u k u in L 1 ( Ω ) implies, up to a subsequence, convergence u k ( x ) u ( x ) almost everywhere in Ω. Hence, by the Lebesgue Theorem, we have

Ω ( u k u ) φ d x 0 , φ L 1 ( Ω ) ,

that is u k u in L ( Ω ) . Since this conclusion is true for any weakly-∗ convergent subsequence of { u k } k N , it follows that u is the weak-∗ limit for the whole sequence { u k } k N . □

Proposition 3.3 A ad is a sequentially compact subset of L r ( Ω ) for any r [ 1 , + ) , and it is a sequentially weakly-∗ compact subset of L ( Ω ) .

Proof Let { u k } k N be any sequence of A ad . Then { u k } k N is bounded in B V ( Ω ) L ( Ω ) . As a result, the statement immediately follows from Propositions 3.2 and 2.2(iii). □

Proposition 3.4For every f W 1 , q ( Ω ) the set Ξ is sequentially compact, i.e. for each sequence { ( u k , y k ) Ξ } k N it can be found a subsequence { ( u k n , y k n ) Ξ } n N such that u k n u 0 in L 1 ( Ω ) , y k n y 0 in W 0 1 , p ( Ω ) , where ( u 0 , y 0 ) Ξ , that is, y 0 is a weak solution to the Dirichlet boundary value problem (3.2) with u = u 0 .

Proof Let { ( u k , y k ) } k N Ξ 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 { ( u k , y k ) } k N and a pair ( u , y ) A ad × W 0 1 , p ( Ω ) such that

y k y  weakly in  W 0 1 , p ( Ω ) , (3.5)

u k u  strongly in  L r ( Ω ) , r [ 1 , + )  and weakly-  in  L ( Ω ) . (3.6)

Taking into account the inequality (see [34])

( | ξ | p 2 ξ | η | p 2 η , ξ η ) R N 2 2 p | ξ η | p ξ , η R N ,

and definition of the class of admissible controls A ad , we conclude: there exists a constant C > 0 independent of k N such that

C Ω | y k y | p d x Ω u k ( | y k | p 2 y k | y | p 2 y , y k y ) R N d x = f , y k y W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) + Ω ( u u k ) ( | y | p 2 y , y k y ) R N d x Ω u ( | y | p 2 y , y k y ) R N d x = I 1 + I 2 I 3 . (3.7)

Since u β , u | y | p 2 y L q ( Ω ; R N ) , and y k y in L p ( Ω ; R N ) , it follows that

I 1 : = f , y k y W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) by (3.5) 0 as  k , I 2 : = Ω u ( | y | p 2 y , y k y ) R N d x by (3.5) 0 as  k , ( u u k ) | y | p 2 y by (3.6) 0  strongly in  L q ( Ω ; R N ) , and, therefore , I 2 : = Ω ( u u k ) ( | y | p 2 y , y k y ) R N d x by (3.5) 0 as  k .

Hence, passing to the limit in (3.7) as k , we arrive at a conclusion (see [26]):

y k y  strongly in  W 0 1 , p ( Ω ) ,  and  | y k | p 2 y k | y | p 2 y  in  L q ( Ω ; R N ) .

As a result, we finally have

f , v W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = lim k Ω u k ( x ) | y k | p 2 ( y k , v ) R N d x = Ω u ( x ) | y | p 2 ( y , v ) R N d x , v W 0 1 , p ( Ω ) ,

that is, the limit pair ( u , y ) 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

( u opt , y opt ) Ξ [ B V ( Ω ) L ( Ω ) ] × W 0 1 , p ( Ω ) , I ( u opt , y opt ) = inf ( u , y ) Ξ I ( u , y ) .

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 f W 1 , q ( Ω ) the mapping u y ( u ) is not Fréchet differentiable on the class of admissible controls, in general, and the class A ad 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 ψ ε to an optimal solution y 0 W 0 1 , p ( Ω ) 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:

Λ ( u , y , μ ) = I ( u , y ) + a u ( y , μ ) f , μ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) , (4.1)

where μ W 0 1 , p ( Ω ) is a Lagrange multiplier and

a u ( y , μ ) = Δ p ( u , y ) , μ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = Ω u ( x ) ( | y | p 2 y , μ ) R N d x .

For given h W 0 1 , p ( Ω ) and θ [ 0 , 1 ] , let us consider the following sets:

Ω 0 = { x Ω : | y | > 0 } , Ω 0 , θ = { x Ω : | y + θ h | > 0 } .

Clearly, we cannot claim that χ Ω 0 , θ χ Ω 0 in L r ( Ω ) for some 1 r < , because convergence of the sequence { y + θ h } θ + 0 to ∇y does not imply, in general, the χ-convergence of subsets { Ω 0 , θ } θ + 0 to Ω 0 as θ + 0 [[15], p.218]. Indeed, let h W 0 1 , p ( Ω ) be such that | h ( x ) | > 0 almost everywhere in Ω and y = 0 . Then Ω 0 = whereas Ω 0 , θ = Ω for all positive θ small enough. Hence, in this case the convergence χ Ω 0 , θ χ Ω 0 fails. In view of this, we make use of the following notion (see [23]).

Definition 4.1 We say that an element y W 0 1 , p ( Ω ) is a regular point for the Lagrangian (4.1) if for each v W 0 1 , p ( Ω ) the direction h = v y is non-degenerate in the following sense:

χ Ω 0 , θ χ Ω 0 in  L 1 ( Ω ) . (4.2)

We have the following result.

Proposition 4.2Assume thatyis an element of W 0 1 , p ( Ω ) such that the set

S 0 = { x Ω : | y ( x ) | = 0 } (4.3)

has zero Lebesgue measure. Then y W 0 1 , p ( Ω ) is a regular point of the Lagrangian (4.1) in the sense of Definition 4.1.

Proof Let h be a given element of W 0 1 , p ( Ω ) . If x Ω 0 , then, by definition, | y ( x ) | > 0 . Thus, there is a value θ 0 ( 0 , 1 ] such that | y ( x ) + θ h ( x ) | > 0 for all θ [ 0 , θ 0 ] . It is worth to note that this pointwise inequality makes a sense if only x Ω is a Lebesgue point of both ∇y and ∇h. However, the Lebesgue Differentiation Theorem states that, given any g L 1 ( Ω ) , almost every x Ω is a Lebesgue point. Hence, almost all Lebesgue points of ∇y are the Lebesgue points of y + θ h for θ small enough, and χ Ω 0 , θ ( x ) = χ Ω 0 ( x ) = 1 for all θ [ 0 , θ 0 ] . Since the set S 0 has zero Lebesgue measure, it follows that | Ω Ω 0 | = 0 and | Ω Ω 0 , θ | = 0 for θ [ 0 , 1 ] small enough. Therefore, χ Ω 0 , θ χ Ω 0 almost everywhere in Ω and hence, χ Ω 0 , θ χ Ω 0 strongly in L 1 ( Ω ) . □

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 S 0 : = { x Ω : y = 0 } 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 W 0 1 , p ( Ω ) are two regular points of the functional Λ ( u , y , λ ) , then there exists a positive number α R ( α 0 ) such that each point of the segment [ y , α v ] = { y + t ( α v y ) : t [ 0 , 1 ] } W 0 1 , p ( Ω ) is also regular for Λ ( u , y , λ ) .

We are now ready to study the differentiability properties of the Lagrangian Λ ( u , y , λ ) . We begin with the following result.

Lemma 4.3Let u A ad be a given element, and let y W 0 1 , p ( Ω ) be a regular point of the Lagrangian (4.1). Then the mapping

W 0 1 , p ( Ω ) y Δ p ( u , y ) = div ( u ( x ) | y | p 2 y ) W 1 , q ( Ω ) , p 2

is Gâteaux differentiable atyand its Gâteaux derivative

( Δ p ( u , y ) ) G L ( W 0 1 , p ( Ω ) , W 1 , q ( Ω ) )

exists and takes the form

( Δ p ( u , y ) ) G [ h ] = div ( u ( x ) | y | p 2 h ) ( p 2 ) div ( u ( x ) | y | p 4 ( y , h ) R N y ) . (4.4)

Proof Let y W 0 1 , p ( Ω ) be a regular point for the Lagrangian (4.1) and let h W 0 1 , p ( Ω ) be an arbitrary distribution. Following the definition of the Gâteaux derivative, we have to deduce the following equality:

lim λ + 0 Δ p ( u , y + λ h ) Δ p ( u , y ) λ ( p 2 ) div ( u ( x ) | y | p 4 ( y , h ) R N y ) div ( u | y | p 2 h ) L q ( Ω ; R N ) = 0 .

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

g ( λ ) : = | y + λ h | p 2 ( y + λ h )

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

| g ( λ ) g ( 0 ) | λ | g ( θ ) | , θ ( 0 , λ ) ,

where g ( 0 ) = | y | p 2 y and

g ( θ ) = | y + θ h | p 2 h + ( p 2 ) | y + θ h | p 2 ( θ | h | 2 + ( y , h ) R N ) ( y + θ h ) 1 | y + θ h | 2 = | y + θ h | p 2 h + ( p 2 ) | y + θ h | p 2 ( y + θ h , h ) R N | y + θ h | y + θ h | y + θ h | .

Let δ > 0 be an arbitrary value. Let us consider the following decomposition:

Ω = S 0 Ω δ Ω δ ′′ ,

where the set S 0 is defined by (4.3), and Ω δ and Ω δ ′′ are measurable subsets of Ω such that

Ω δ = { x Ω : | y ( x ) | δ } , Ω δ ′′ = { x Ω : 0 < | y ( x ) | < δ } .

Following [[34], p.598] (see also [38]), we have: for any ε > 0 there exists a positive value δ 0 > 0 such that

g ( θ ) λ ( p 2 ) | y | p 4 ( y , h ) R N y | y | p 2 h L q ( Ω δ ; R N ) < ε 2 , (4.5)

g ( θ ) λ ( p 2 ) | y | p 4 ( y , h ) R N y | y | p 2 h L q ( Ω δ ′′ ; R N ) < ε 2 (4.6)

for all δ ( 0 , δ 0 ) , θ ( 0 , λ ) , and λ > 0 small enough. Since | S 0 | = 0 by the initial assumptions, it follows from (4.5)-(4.6) that the vector-valued function | y | p 2 y is Gâteaux differentiable. Hence, the operator Δ p ( u , y ) = div ( u ( x ) | y | p 2 y ) is Gâteaux differentiable for any regular point y W 0 1 , p ( Ω ) and for any admissible control u A ad , and its Gâteaux derivative takes the form (4.4). □

Since Gâteaux differentiability of the operator y Δ p ( u , y ) implies the existence of Gâteaux derivative for the functional

φ ( y ) = Δ p ( u , y ) , μ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = Ω u ( x ) ( | y | p 2 y , μ ) R N d x , φ : W 0 1 , p ( Ω ) R

such that

φ G ( y ) , h W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = ( Δ p ( u , y ) ) G [ h ] , μ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) , μ W 0 1 , p ( Ω ) ,

we arrive at the following obvious consequence of Lemma 4.3.

Corollary 4.4Let u A ad and λ W 0 1 , p ( Ω ) be given elements, and let y W 0 1 , p ( Ω ) be a regular point of the Lagrangian (4.1). Then the mapping

W 0 1 , p ( Ω ) y Λ ( u , y , μ ) = I ( u , y ) + a u ( y , μ ) f , μ W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) R

is Gâteaux differentiable atyand its Gâteaux derivative,

Λ G ( u , y , μ ) W 1 , q ( Ω ) ,

exists and takes the form

Λ G ( u , y , μ ) , h W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = p Ω | y y d | p 2 ( y y d , h ) R N d x + Ω ( u ( x ) | y | p 2 μ , h ) R N d x + ( p 2 ) Ω u ( x ) | y | p 4 ( y , μ ) R N ( y , h ) R N d x . (4.7)

Remark 4.2 In view of the equality

( y , μ ) R N y = [ y y ] μ ,

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

( p 2 ) Ω u ( x ) | y | p 4 ( [ y y ] μ , h ) R N d x .

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

Lemma 4.5Let u A ad , y W 0 1 , p ( Ω ) , and v W 0 1 , p ( Ω ) be given distributions. Assume that each point of the segment [ y , v ] = { y + α ( v y ) : α [ 0 , 1 ] } W 0 1 , p ( Ω ) is regular for the mapping v Λ ( u , v , μ ) . Then there exists a positive value ε ( 0 , 1 ) such that

Λ ( u , v , μ ) Λ ( u , y , μ ) = Λ G ( u , y + ε h , μ ) , h W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) = p Ω | y + ε h y d | p 2 ( y + ε h y d , h ) R N d x + Ω u ( x ) | y + ε h | p 2 ( μ , h ) R N d x + ( p 2 ) Ω u ( x ) | y + ε h | p 4 × ( [ ( y + ε h ) ( y + ε h ) ] μ , h ) R N d x (4.8)

with h = v y .

Proof For given u, μ, y d , y, and v, let us consider the scalar function φ ( t ) = Λ ( u , y + t ( v y ) , μ ) . Since by Corollary 4.4, the functional Λ ( u , , μ ) is Gâteaux differentiable at each point of the segment [ y , v ] , it follows that the function φ = φ ( t ) is differentiable on ( 0 , 1 ) and

φ ( t ) = Λ G ( u , y + t ( v y ) , λ ) , v y W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) , t ( 0 , 1 ) .

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

φ ( 1 ) φ ( 0 ) = φ ( ε ) for some  ε ( 0 , 1 ) .

 □

5 Formalism of the quasi-adjoint technique

We begin with the following assumption:

(H1) The distribution f W 1 , q ( Ω ) is such that, for each admissible control u A ad , the corresponding weak solution y = y ( u ) of the nonlinear Dirichlet boundary value problem (2.3) is a regular point of the mapping y Λ ( u , y , λ ) .

Let ( u 0 , y 0 ) Ξ be an optimal pair for problem (3.1)-(3.3). Then

Δ Λ = Λ ( u , y , λ ) Λ ( u 0 , y 0 , λ ) 0 , ( u , y ) Ξ , λ W 0 1 , p ( Ω ) . (5.1)

Hence, splitting it in two terms, we obtain

Λ ( u , y , λ ) Λ ( u 0 , y 0 , λ ) = Λ ( u , y , λ ) Λ ( u , y 0 , λ ) + Λ ( u , y 0 , λ ) Λ ( u 0 , y 0 , λ ) = Δ y Λ ( u , y 0 , λ ) + Λ ( u u 0 , y 0 , λ ) 0 , (5.2)

for all λ W 0 1 , p ( Ω ) and u A ad such that ( u u 0 ) A ad .

Since the set of admissible controls A ad L 1 ( Ω ) has an empty topological interior, we justify the choice of perturbation for an optimal control as follows:

u θ : = u 0 + θ ( u ˆ u 0 ) ,

where ( u ˆ , y ˆ ) Ξ is an arbitrary admissible pair, and θ [ 0 , 1 ] . Then due to Hypothesis (H1) and Remark 4.1, we can suppose that each point of the segment [ y 0 , y θ ] W 0 1 , p ( Ω ) is regular for the mapping v Λ ( u , v , λ ) , where y θ : = y ( u θ ) = y ( u 0 + θ ( u ˆ u 0 ) ) is the corresponding solution of boundary value problem (3.2)-(3.3). Hence, by Lemma 4.5, there exists a value ε θ ( 0 , 1 ) such that condition (5.1) can be represented as follows:

Δ Λ = Λ ( u θ , y θ , λ ) Λ ( u 0 , y 0 , λ ) = Λ G ( u θ , y 0 + ε θ ( y θ y 0 ) , λ ) , y θ y 0 W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) + Λ ( u θ u 0 , y 0 , λ ) = Λ G ( u θ , y 0 + ε θ ( y θ y 0 ) , λ ) , y θ y 0 W 1 , q ( Ω ) ; W 0 1 , p ( Ω ) + Λ ( θ ( u ˆ u 0 ) , y 0 , λ ) 0 . (5.3)

Using (4.7), we obtain

Δ Λ = p Ω | y ε θ , θ y d | p 2 ( y ε θ , θ y d , y θ y 0 ) R N d x + Ω u θ | y ε θ , θ | p 2 ( λ , y θ y 0 ) R N d x + ( p 2 ) Ω u θ | y ε θ , θ | p 4 ( [ y ε θ , θ y ε θ , θ ] λ , y θ y 0 ) R N d x + θ Ω ( u ˆ u 0 ) ( | y 0 | p 2 y 0 , λ ) R N d x 0 , u ˆ A ad , (5.4)

where y ε θ , θ = y 0 + ε θ ( y θ y 0 ) .

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 θ [ 0 , 1 ] and u ˆ A ad , a distribution ψ ε θ , θ is the quasi-adjoint state to y 0 W 0 1 , p ( Ω ) if ψ ε θ , θ satisfies the following integral identity:

Ω u θ | y ε θ , θ | p 2 ( [ I + ( p 2 ) y ε θ , θ | y ε θ , θ | y ε θ , θ | y ε θ , θ | ] ψ ε θ , θ , φ ) R N d x + p Ω | y ε θ , θ y d | p 2 ( y ε θ , θ y d , φ ) R N d x = 0 , φ W 0 1 , p ( Ω ) (5.5)

or in the operator form

div ( u θ | y ε θ , θ | p 2 [ I + ( p 2 ) y ε θ , θ | y ε θ , θ | y ε θ , θ | y ε θ , θ | ] ψ ε θ , θ ) = p div ( | y ε θ , θ y d | p 2 ( y ε θ , θ y d ) ) in  D ( Ω ) . (5.6)

Here, I L ( R N ; R N ) is the identity matrix, y θ : = y ( u θ ) = y ( u 0 + θ ( u ˆ u 0 ) ) is the solution of problem (3.2)-(3.3), y ε θ , θ = y 0 + ε θ ( y θ y 0 ) , and ε θ = ε ( u θ ) ( 0 , 1 ) is a constant coming from equality (5.3).

Let us assume, for a moment, that quasi-adjoint state ψ ε θ , θ to y 0 W 0 1 , p ( Ω ) is a distribution of W 0 1 , p ( Ω ) . Then we can define the element λ in (5.4) as the quasi-adjoint state, that is, we can set λ = ψ ε θ , θ . As a result, the increment of Lagrangian (5.4) can be simplified to the form

Ω ( u ˆ u 0 ) ( | y 0 | p 2 y 0 , ψ ε θ , θ ) R N d x 0 , u ˆ A ad . (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 θ + 0 , and to show that the sequence of quasi-adjoint states { ψ ε , θ } θ 0 is defined in a unique way through relation (5.5) and it is compact with respect to the weak topology of W 0 1 , p ( Ω ) .

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 u k , u A ad and u k u strongly in L ( Ω ) . Then, for the corresponding solutions of boundary value problem (3.2)-(3.3), we have strong convergence y k = y ( u k ) y = y ( u ) in W 0 1 , p ( Ω ) .

Since, by the initial suppositions, u θ u 0 in L ( Ω ) as θ 0 , we immediately arrive at the following consequence of Lemma 6.1.

Corollary 6.2

(i) y θ y 0 in W 0 1 , p ( Ω ) as θ 0 ;

(ii) y ε θ , θ y 0 in W 0 1 , p ( Ω ) as θ 0 .

Further, we note that

(A1) if r ( 1 , p ] , where p 2 , and f , g L p ( Ω ) , then

| | f ( x ) | r | g ( x ) | r | r ( | f ( x ) | + | g ( x ) | ) r 1 | f ( x ) g ( x ) | , a.e. in  Ω , (6.1)

and, hence, by the Hölder Inequality,

| f L r ( Ω ) r g L r ( Ω ) r | = | Ω | f | r d x Ω | g | r d x | Ω | | f | r | g | r | d x f g L p ( Ω ) ( Ω ( | f | + | g | ) ( r 1 ) p / ( p 1 ) d x ) ( p 1 ) / p f g L p ( Ω ) | f | + | g | L p ( Ω ) r 1 | Ω | ( p r ) / p . (6.2)

(A2) If r [ 0 , 1 ] and f , g L p ( Ω ) then

| | f ( x ) | r | g ( x ) | r | | f ( x ) g ( x ) | r , a.e. in  Ω , (6.3)

and, therefore,

| f L r ( Ω ) r g L r ( Ω ) r | Ω | f g | r d x f g L p ( Ω ) r | Ω | ( p r ) / p . (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 r [ 0 , p ] with p 2 , we have | y ε θ , θ | r | y 0 | r in L 1 ( Ω ) as θ 0 .

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

div ( ρ θ A θ ψ ε θ , θ ) = f θ in  Ω , (6.5)

where

ρ θ : = | y ε θ , θ | p 2 , (6.6)

A θ : = u θ [ I + C θ ] = u θ [ I + ( p 2 ) y ε θ , θ | y ε θ , θ | y ε θ , θ | y ε θ , θ | ] , (6.7)

f θ : = p div ( | y ε θ , θ y d | p 2 ( y ε θ , θ y d ) ) . (6.8)

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

Proposition 6.4Let ( u 0 , y 0 ) Ξ be an optimal pair for problem (3.1)-(3.3). Then, for any θ [ 0 , 1 ] and u ˆ A ad , we have:

(1) A θ L ( Ω ; S sym N ) , where S sym N denotes the set of all N × N symmetric matrices, which are obviously determined by N ( N + 1 ) / 2 scalars.

(2) α | ξ | 2 ( ξ , A θ ξ ) R N β [ 1 + ( p 2 ) 2 N 1 ] | ξ | 2 for all ξ R N .

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 ad and the following chain of estimates:

α | ξ | 2 ( ξ , u θ I ξ ) R N ( ξ , u θ I ξ ) R N + ( ξ , u θ C θ ξ ) R N = ( ξ , u θ I ξ ) R N + ( p 2 ) u θ ( y ε θ , θ | y ε θ , θ | , ξ ) R N 2 β | ξ | 2 + ( p 2 ) β ( ξ 1 + + ξ N ) 2 β [ 1 + ( p 2 ) 2 N 1 ] | ξ | 2 .

 □

Proposition 6.5

(a) ρ θ : = | y ε θ , θ | p 2 | y 0 | p 2 = : ρ 0 in L 1 ( Ω ) as θ 0 ;

(b) A θ A 0 in L r ( Ω ; S sym N ) for any r [ 1 , + ) and A θ A 0 in L ( Ω ; S sym N ) as θ 0 ;

(c) ρ θ A θ ρ 0 A 0 = u 0 [ I + ( p 2 ) y 0 | y 0 | y 0 | y 0 | ] in L 1 ( Ω ; R N ) as θ 0 .

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 θ [ 0 , 1 ] and u ˆ A ad , ln | y ε θ , θ | L 2 ( Ω ; R N ) . Then each elementψof W 0 1 , p ( Ω ) can be represented in a unique way as follows:

ψ = | y ε θ , θ | ( 2 p ) / 2 z θ , where  z θ W 0 1 , 1 ( Ω ) L 2 ( Ω ) . (6.9)

Proof Let us fix an element ψ W 0 1 , p ( Ω ) . Then for z θ : = | y ε θ , θ | ( p 2 ) / 2 ψ , by the Hölder Inequality with p = p / 2 and q = p / ( p 2 ) , we have

z θ L 2 ( Ω ) 2 = Ω | y ε θ , θ | p 2 ψ 2 d x y ε θ , θ L p ( Ω ; R N ) p 2 ψ L p ( Ω ) 2 C y ε θ , θ W 0 1 , p ( Ω ) p 2 ψ W 0 1 , p ( Ω ) 2 < ,

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

z θ = ( 1 2 ρ ψ ln ρ + ρ ψ ) | ρ = | y ε θ , θ | p 2 ,

and applying the Hölder Inequality with exponents p = 2 ( p 1 ) / ( p 2 ) and q = 2 ( p 1 ) / p for the first term and with p = p / 2 and q = p / ( p 2 ) for the second one, we get

z θ L 1 ( Ω ; R N ) p 2 2 Ω | y ε θ , θ | ( p 2 ) / 2 | ψ | | ln | y ε θ , θ | | d x + | Ω | 1 / 2 ( Ω | y ε θ , θ | p 2 | ψ | 2 d x ) 1 / 2 p 2 2 ψ L p ( Ω ) ( Ω | y ε θ , θ | ( p 2 ) p 2 ( p 1 ) | ln | y ε θ , θ | | p p 1 d x ) ( p 1 ) / p + | Ω | 1 / 2 y ε θ , θ L p ( Ω ; R N ) ( p 2 ) / 2 ψ L p ( Ω ; R N ) p 2 2 ψ L p ( Ω ) y ε θ , θ L p ( Ω ; R N ) ( p 2 ) / 2 ln | y ε θ , θ | L 2 ( Ω ; R N ) + | Ω | 1 / 2 y ε θ , θ W 0 1 , p ( Ω ) ( p 2 ) / 2 ψ W 0 1 , p ( Ω ) ( p 2 2 C ln | y ε θ , θ | L 2 ( Ω ; R N ) + | Ω | 1 / 2 ) y ε θ , θ W 0 1 , p ( Ω ) ( p 2 ) / 2 ψ W 0 1 , p ( Ω ) < .

Thus, z θ W 1 , 1 ( Ω ) L 2 ( Ω )