Open Access Research

A generalization on the solvability of integral geometry problems along plane curves

Zekeriya Ustaoglu

Author Affiliations

Department of Mathematics, Bulent Ecevit University, Zonguldak, 67100, Turkey

Boundary Value Problems 2013, 2013:202  doi:10.1186/1687-2770-2013-202


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/202


Received:2 August 2013
Accepted:19 August 2013
Published:8 September 2013

© 2013 Ustaoglu; 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 cited.

Abstract

This paper is concerned with a general condition for the solvability of integral geometry problems along the plane curves of given curvatures. As two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. By using some extension of the class of unknown functions, the proofs are based on the solvabilities of equivalent inverse problems for transport-like equation.

MSC: 35R30, 53C65, 65N30.

Keywords:
integral geometry problem; inverse problem; Galerkin method; transport-like equation

1 Introduction

The problems of integral geometry are to determine a function, given (weighted) integrals of this function over a family of manifolds, and there has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is unity, there are interesting results in the plane case when a family of curves is regular or in the case of a family of straight lines with arbitrary regular attenuation [[1], Chapter 7]. It is assumed that the basis of the integral geometry problems is the Radon transform [2]. The Radon transform R integrates a function f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M1">View MathML</a> over hyperplanes. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M2">View MathML</a> be the hyperplane perpendicular to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M3">View MathML</a> (unit sphere) with signed distance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M4">View MathML</a> from the origin, and the Radon transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M5">View MathML</a> is defined as the integral of f over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M6">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M7">View MathML</a>

(see [[3], Chapter 2]).

The problems of integral geometry have important applications in imaging and provide the mathematical background of tomography, where the main goal is to recover the interior structure of a nontransparent object using external measurements. The object under investigation is exposed to radiation at different angles, and the radiation parameters are measured at the points of observation. The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals, and there are many applications related with computerized tomography: medical imaging, geophysics, diagnostic radiology, astronomy, seismology, radar and many other fields (see, e.g., [4]).

From the applied point of view, the importance of integral geometry problem over a family of straight lines in the plane is indicated in [5], where the problem models X-rays, and applicable to the problems of radiology and radiotherapy. Because of their many practical applications, a considerable attention has been devoted to other family of curves in the plane as well as straight lines. Invertibility of the Radon transforms on some families of curves in the plane is given with explicit inversion formulas via circular harmonic decomposition in [6] and for the explicit inversion formulas of the attenuated Radon transform, see, e.g., [7,8]. Note that the circle is the simplest non-trivial curve in the plane next to the straight line, and the representation of a function by its circular Radon transform also arises in applications. In [9], invertibility of the Radon transforms over all translations of a circle of fixed radius and circles of varying radius centered on a fixed circle is considered, where the proofs require microlocal analysis of the Radon transforms and a microlocal Holmgren theorem. In [10], some existence and uniqueness results on recovering a function from its circular Radon transform with partial data are presented and the relations to applications in medical imaging are described. There are several other ways related to the selection of a family of curves, such as circles of varying radius centered on a straight line or a fixed curve, circles passing through a fixed point, along paths that are not on the zero sets of harmonic polynomials, circular arcs having a chord of fixed length rotating around its middle point etc., which are meaningful in applications on thermo-acoustic and photoacoustic tomography, synthetic aperture radar, Compton scattering tomography, ultrasound tomography etc. (see, e.g., [10-12] and the references therein).

In fact, since the seminal work of Radon [2], the various integral geometry problems with numerous applications have been considered in several important aspects which are not mentioned here, but for a comprehensive list, see, e.g., [13-16] and the references therein. Furthermore, the problems of integral geometry and inverse problems for transport equations are interrelated and the latter are also of great importance in theory and applications; see, e.g., [17-20] and for the derivation and applications of transport equations, see, e.g., [21-23].

In this paper, a general condition for the solvability of integral geometry problems along plane curves of given curvatures is presented and its relation with some previous results is indicated. Moreover, as two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. Since the curvature of a circle is defined to be the reciprocal of the radius of the circle, in the former case the curvature is a constant, while in the latter one the curvature depends on the point and the direction. To investigate the solvability of the integral geometry problem (IGP) given in Section 2.1, which is overdetermined since the underlying operator of the IGP is compact and its inverse operator is unbounded (see Section 2.2), it is reduced to an equivalent overdetermined inverse problem for a transport-like equation, and then, with the use of a similar method which was proposed in [24] (see also [[13], Chapter 1]), on using some extension of the class of unknown functions, this inverse problem is replaced by a determined one. Thus, the solvability of IGP is proved via the solvability of an inverse problem for a transport-like equation. The above mentioned method on the solvability was also previously utilized in [25-27] for IGP along some family of plane curves of given curvatures and straight lines, in [28] for IGP along geodesics and in [29] for IGP along the family of curves whose curvatures are given by the Christoffel symbols. Here, the presented general condition for the solvability covers those of [25-28], in the manner indicated by Remark 1 in Section 3.1. Moreover, these previous solvability conditions do not hold for IGP along the above given two families of circles, and this is the main importance and motivation of this study.

In Section 2, IGP and its reduction to the equivalent inverse problem for a transport-like equation, the method to overcome the difficulty on investigating the solvability arising from overdeterminacy of these problems are presented and some definitions and notations which will be used throughout the paper are introduced. Section 3 is devoted to the statements of main results, and finally in Section 4, proofs of the results are given.

The investigation of approximate solutions of the concerned integral geometry problems is beyond the scope of this paper, but similar procedures as in [26] can be carried out by using the Galerkin method or the finite difference method.

2 Statement of the problem and overdeterminacy

2.1 Statement of the problem

Let D be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M8">View MathML</a>. It is assumed that in D, a family of regular curves is given by curvature <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M9">View MathML</a> which is the curvature of the curve passing from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M10">View MathML</a> in the direction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M11">View MathML</a>, and there exists a unique sufficiently smooth curve of this family which is passing from any point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M12">View MathML</a> in the arbitrary direction ν, with the endpoints on the boundary of D. Suppose that the lengths of these curves in D are bounded above by the same constant. Let us denote the family of these curves by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13">View MathML</a>. IGP is stated below.

IGPDetermine a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14">View MathML</a>in the domainDfrom the integrals ofλalong the curves of a given family of curves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M16">View MathML</a> vanishes outside D, and let us introduce an auxiliary function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M17">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18">View MathML</a> is a part of the curve that belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M19">View MathML</a>, with one end of it being the point x and the other one on ∂D, and is the arc length element along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18">View MathML</a>.

Investigating the uniqueness of a solution of a problem of integral geometry by reducing it to the equivalent inverse problem for a differential equation was first carried out in [30]. Similar reduction is demonstrated for IGP formulated below.

Differentiating (1) in the direction ν at x, we obtain the following transport-like equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M21">View MathML</a>

(2)

From (1), u is 2π-periodic with respect to φ, and since the integrals of λ along the curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M22">View MathML</a> are known, u is known on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M24">View MathML</a>

(3)

(see [30] and [[13], p.11]). So, we have the following inverse problem.

Problem 1 Determine a pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a> from the transport-like equation (2) provided that the function K is known and u satisfies conditions (3).

2.2 Overdeterminacy

Generally, in the theory of integral geometry, reconstruction of a function of n variables from a function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M26">View MathML</a> variables is said to be an overdetermined problem of integral geometry (see, e.g., [[14], Chapter 5]), and in the theory of inverse problems, overdetermination usually means that the number m of independent variables in the data exceeds the number n of independent variables in the unknown target function (see, e.g., [[31], Section 1.3]). However, since the data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27">View MathML</a> given on the two-dimensional surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14">View MathML</a> is a function of two variables, these are not the cases for IGP or Problem 1. Here, since the operator given in (1) (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M18">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M31">View MathML</a>) is compact, its inverse is unbounded, and therefore it is not possible to prove a general existence result. So, IGP and Problem 1 are called overdetermined in this sense. Hence, because of the overdeterminacy, the initial data for these problems should not be arbitrary and satisfy some ‘solvability conditions’ (see [[13], p.4] and [[15], p.18, Theorem 1.4]) which are difficult to establish. It should be noted that the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27">View MathML</a> for which IGP is solvable is not everywhere dense in any of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M35">View MathML</a>. Moreover, the data in problems of integral geometry are of quasianalytic character, i.e., their values specified in a domain of the Lebesgue measure can be as small as desired, determine their values in an essentially larger domain (see [[32], Chapter 6, Section 17] and [[33], Chapter 6, Section 1]). In particular, this implies that it is impossible to avoid overdeterminacy of the problem by specifying the data on a part of the boundary rather than on the whole boundary. Even if it were possible to find the solvability conditions for the mentioned overdetermined problems, since the real data usually have some errors in practice, and thus fall out of the data class for which the existence of a solution is established, it appears that these conditions would not always be satisfactory in applications. Therefore, to prove the existence results, such special conditions on the data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M27">View MathML</a> have to be posed.

Let us propose the procedure for establishing the solvability of IGP. Assume that the unknown function λ in IGP depends not only upon the space variables x, but also upon the direction φ in some special manner, i.e., consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37">View MathML</a>, where this dependence upon φ is impossible to be arbitrary, for in the opposite case the problem would be underdetermined and the examples on the nonuniqueness of a solution can be easily constructed. Herein the special dependence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37">View MathML</a> upon the direction means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37">View MathML</a> satisfies a certain differential equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40">View MathML</a>, where the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M41">View MathML</a> is given in Section 2.3) with the following properties:

(1) The IGP or Problem 1 with the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37">View MathML</a> becomes a determined one.

(2) The sufficiently smooth functions λ depending only on x satisfy this equation.

Suppose that a differential equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M37">View MathML</a> satisfying properties (1) and (2) has been found and that a priori the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44">View MathML</a>, which represents the exact data of IGP related to a function λ depending only on x, is known. Then, utilizing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44">View MathML</a>, a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M46">View MathML</a> to IGP can be constructed. By uniqueness of a solution, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M14">View MathML</a> coincide. At the same time, knowing the approximate data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M50">View MathML</a>, an approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M51">View MathML</a> can be constructed such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M52">View MathML</a>. Recall that if λ depends only on x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49">View MathML</a> does not satisfy the ‘solvability conditions’, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M54">View MathML</a> depending only x does not exist. Here the data are specified on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M56">View MathML</a> is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M49">View MathML</a>. In other words, a regularizing procedure is constructed for the IGP.

In general, the equation with the properties (1) and (2) for the same problem is not uniquely defined. Hence, the class of unknown functions λ extends so that IGP for this class becomes a determined problem and all sufficiently smooth functions in x belong to it. On using some extension of the class of functions λ, the overdetermined Problem 1 is replaced by a determined one (Problem 2 in Section 3.1).

The above method of solvability of the IGP or Problem 1 leads to the Dirichlet-type problem with conditions (3) for the third-order equation of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M59">View MathML</a>. In investigating and proving the solvability of IGP over any regular family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M13">View MathML</a> of curves with curvature K, since the quadratic form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M61">View MathML</a> in (14) is required to be positive definite, the construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M41">View MathML</a> is important and to be able to this, in Theorem 1 condition (6) is given.

2.3 Definitions and notations

In this section, some notations are given based on [13]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63">View MathML</a> with the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M66">View MathML</a> be the closure of Ω. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M67">View MathML</a> we denote a scalar product of functions u and v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M69">View MathML</a> the set of all functions defined in Ω which have continuous partial derivatives of order up to all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M70">View MathML</a>, whose supports are compact subsets of Ω. For a differential expression A, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M71">View MathML</a> we denote the conjugate of A in the sense of Lagrange. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M72">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M73">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M74">View MathML</a> denote the set of real-valued functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M75">View MathML</a> that are 2π-periodic with respect to φ in the domain Ω, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M76">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M77">View MathML</a> are nonnegative integers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M78">View MathML</a>.

The proof of Theorem 1 involves energy-like estimates and the Galerkin method (see, e.g., [[34], Chapter 5, Section 2.3], [[35], Chapter 7]), and therefore some class of functions are introduced below. In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M79">View MathML</a>, let us introduce the scalar product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M81">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M82">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M84">View MathML</a> be the completions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M74">View MathML</a> with respect to the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M87">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M88">View MathML</a>), respectively (for the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M89">View MathML</a>, see, e.g., [34,36]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M90">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M92">View MathML</a> be the completions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M93">View MathML</a> with respect to the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M87">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M88">View MathML</a>). Let us take a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M97">View MathML</a> which is complete and orthonormal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a>, then we may assume that the linear span of this set is everywhere dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M91">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M100">View MathML</a> is separable, there exists a countable set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M101">View MathML</a> which is everywhere dense in this space and this set up can be extended to a set which is everywhere dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a>. Orthonormalizing the latter in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M103">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M104">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M105">View MathML</a> the orthogonal projector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M106">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M107">View MathML</a> which is the linear span of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M108">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M109">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M110">View MathML</a> and it can be easily verified that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M111">View MathML</a>

The existence of the function g in the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M112">View MathML</a> leads to a generalization on conditions for the solvability of integral geometry problems. In Theorem 1, it is shown that if there exists a function g satisfying condition (6) which depends on the curvature K and the domain D, the solvability holds.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M114">View MathML</a> be the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M115">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M116">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M117">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M118">View MathML</a> in the generalized functions sense, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M119">View MathML</a> holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M120">View MathML</a>. Take a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M121">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122">View MathML</a> there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M123">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M124">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M126">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M127">View MathML</a>. If we denote the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M93">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M129">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M130">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M131">View MathML</a> and it can be shown that the inclusions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M132">View MathML</a> hold.

3 Statements of results

3.1 Solvability of IGP along plane curves

Since Problem 1 is overdetermined, as indicated in Section 2.2, we consider the following determined problem.

Problem 2 Determine a pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a> defined in Ω that satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M134">View MathML</a>

(4)

provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40">View MathML</a>, u is 2π-periodic with respect to φ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M136">View MathML</a> and K are known.

In (4), it is assumed that the unknown function λ depends also on φ and the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40">View MathML</a> holds in the generalized functions sense.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M64">View MathML</a>, then there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M140">View MathML</a> (see [[34], p.130, Theorem 2]) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M141">View MathML</a>, and we can consider the new unknown function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M142">View MathML</a>. Hence from (4) we obtain the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M143">View MathML</a> , where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M145">View MathML</a>. Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M146">View MathML</a> again by u for simplicity, then Problem 2 can be reduced to Problem 3 given below (see [[13], p.20]) and the solvability of the former follows from that of the latter and does not depend on the choice of G.

Problem 3 Determine a pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a> defined in Ω that satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M148">View MathML</a>

(5)

provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40">View MathML</a>, u is 2π-periodic with respect to φ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M150">View MathML</a>, K and F are known.

The existence, uniqueness and the stability of the solution of Problem 3 are given by the following theorem.

Theorem 1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M152">View MathML</a>exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M153">View MathML</a>

(6)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154">View MathML</a>, then Problem 3 has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M157">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M158">View MathML</a>

(7)

holds, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M56">View MathML</a>depends onKand the Lebesgue measure ofD.

Remark 1 In fact, without being aware of (6), the function g, with the appropriate choices of it, was used previously in [25,26,28]. The convenience of (6) with those of previous solvability results is indicated below.

(i) In [26], when the curvature K is sufficiently smooth and 2π-periodic, the condition for the solvability is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M160">View MathML</a>for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154">View MathML</a>, where this condition holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M162">View MathML</a> in (6).

(ii) In [25], when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M163">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M165">View MathML</a> are sufficiently smooth functions, the condition for the solvability is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M166">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M167">View MathML</a>, which holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M168">View MathML</a> in (6).

(iii) In [28], when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M169">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M171">View MathML</a> are sufficiently smooth 2π-periodic functions, the condition for the solvability is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M172">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M173">View MathML</a>, and this condition holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M174">View MathML</a> in (6).

Note that in none of the above cases, the solvability conditions hold for the IGP along the family of circles of varying radius centered on a fixed circle and the family of curves of constant curvatures, i.e., the family of circles with fixed radius where the curvature is a nonzero constant or the family of straight lines where the curvature is zero. The former cases are investigated in Section 3.2 below and the latter case is considered in [27], where the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M175">View MathML</a> in the expression of Lu will not be present since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M176">View MathML</a> and the proof of solvability is given for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M162">View MathML</a>. In fact, the strict inequality in (6) can be written as a non-strict inequality, with the equality only for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M178">View MathML</a>.

3.2 Solvability of IGP along the family of circles

Since Theorem 1 was given for IGP along a regular family of plane curves for the general case, in this section the given results on the solvabilities of IGP along the family of circles depend on finding an appropriate function g satisfying (6) for the given curvature K and the domain D under the assumptions of Theorem 1.

3.2.1 Solvability of IGP along the family of circles with fixed radius

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180">View MathML</a>. Let us take the family of curves in IGP as the family of circular arcs (the segments of circles inside D with the endpoints on ∂D) with fixed radius r, passing from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M181">View MathML</a> in the direction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M182">View MathML</a> and denote this family of circles by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M183">View MathML</a>. Since the curvature of a circle is defined to be the reciprocal of the radius of this circle, the curvature of the elements of the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M184">View MathML</a> becomes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185">View MathML</a>.

The solvability of IGP along the curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M183">View MathML</a> follows from the following lemma.

Lemma 1Let us definegon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M188">View MathML</a>

(8)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M189">View MathML</a>, then (6) holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185">View MathML</a>.

Remark 2 The above choice of g is not unique, and since the curvature <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M191">View MathML</a> is constant, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M192">View MathML</a>, condition (6) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M193">View MathML</a>

(9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M154">View MathML</a>, which depends on r and the domain D.

3.2.2 Solvability of IGP along the family of circles of varying radius centered on a fixed circle

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M197">View MathML</a> be a fixed disk of radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M198">View MathML</a> and, without any loss of generality, centered at the origin such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199">View MathML</a>, and take the family of curves in IGP as the family of circular arcs (the segments of circles inside D with the endpoints on ∂D) of varying radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M200">View MathML</a> centered on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M201">View MathML</a>, passing from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M181">View MathML</a> in the direction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M203">View MathML</a> and denote this family of circular arcs by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204">View MathML</a>. It can be shown that the radius of the circles of the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204">View MathML</a> is defined by the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M206">View MathML</a>

(10)

and hence the curvature of the elements of the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M208">View MathML</a>

(11)

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187">View MathML</a>. Note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M212">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M213">View MathML</a>.

The solvability of IGP along the curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M204">View MathML</a> follows from Lemma 2 given below.

Lemma 2If the functiongis defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M216">View MathML</a>

(12)

then (6) holds forKdefined in (11).

4 Proof of results

Proof of Theorem 1 First, we will prove the uniqueness of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a> of Problem 3 under the assumptions of the theorem. To this end, it is sufficient to show that the corresponding homogeneous problem has only a trivial solution. Then, by taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M219">View MathML</a>, from (5) we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M220">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M124">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M225">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M127">View MathML</a>. Now we want to decompose the product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M227">View MathML</a> into the sum of a positive definite quadratic form and a divergence form. For this purpose, we have the following identities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M228">View MathML</a>

(13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M229">View MathML</a>

If (6) holds, then the quadratic form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M230">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M233">View MathML</a> is positive definite, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M234">View MathML</a>

(14)

Indeed, we can estimate the terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M236">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M237">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M238">View MathML</a>, and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M239">View MathML</a>

Moreover, whenever (6) holds, for sufficiently close value of ε to 1, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M240">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M241">View MathML</a> in Ω, and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M242">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M243">View MathML</a>.

Since the domain D is bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M244">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M230">View MathML</a> is positive definite, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M247">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M249">View MathML</a> depends on the Lebesgue measure of D and does not depend on k.

Thus, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M250">View MathML</a>, K and g are 2π-periodic with respect to φ, after integrating (13) over Ω, the divergent terms disappear and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M251">View MathML</a>

(15)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122">View MathML</a>, from (15) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M253">View MathML</a>

(16)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M254">View MathML</a>, and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M219">View MathML</a>, from (5) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M256">View MathML</a>. So, the uniqueness part of the proof is completed.

Now we will prove that there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M25">View MathML</a> of Problem 3 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M258">View MathML</a> by means of the following auxiliary problem.

Determineudefined in Ω that satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M259">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M260">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M261">View MathML</a>.

The solution u of problem (17)-(18) will be approximated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M262">View MathML</a>

construction of which is based on finding the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M263">View MathML</a> from the system of linear algebraic equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M264">View MathML</a>

(19)

where the system of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M265">View MathML</a> is taken as indicated in Section 2.3 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M266">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M23">View MathML</a>.

We must show that the solution of system (19) exists and is unique for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151">View MathML</a>. To demonstrate this, let us assume that the homogeneous version of (19), i.e., the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M269">View MathML</a>

has a nonzero solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M270">View MathML</a>. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M271">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M263">View MathML</a>, multiplying the jth equation of the above homogenous system by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M273">View MathML</a> and summing with respect to j from 1 to N, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M274">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M275">View MathML</a>. So, from (15) and (20) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M276">View MathML</a>

and since the quadratic form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M277">View MathML</a> defined in (14) is positive definite and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M278">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M279">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M278">View MathML</a> in Ω. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M281">View MathML</a> is linearly independent and this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M283">View MathML</a>, which contradicts with the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M284">View MathML</a>. So, it is shown that system (19) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M285">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151">View MathML</a>.

Now we estimate the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M287">View MathML</a> of system (19) in terms of F. For this purpose, we multiply both sides of the jth equation of (19) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M288">View MathML</a> and sum the obtained equations with respect to j from 1 to N to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M289">View MathML</a>

(21)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M290">View MathML</a>, applying integration by parts, the right-hand side of (21) can be estimated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M291">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M292">View MathML</a>, and from (15) and (21) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M293">View MathML</a>

(22)

It can be verified that for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M292">View MathML</a>, from (22) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M295">View MathML</a>

(23)

where the constant C is independent of N. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M296">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298">View MathML</a>, and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298">View MathML</a> are Hilbert spaces, it is weakly compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298">View MathML</a>. Therefore, there exists a subsequence, which we again denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M306">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M308">View MathML</a>

holds. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M298">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M312">View MathML</a>. From (23) we have also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M314">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M315">View MathML</a> are bounded and there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303">View MathML</a>, which is again denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M303">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M320">View MathML</a> converge weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M323">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M324">View MathML</a>, respectively. Taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M325">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151">View MathML</a> and applying integration by parts in (19), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M327">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M328">View MathML</a>

Since the linear span of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M329">View MathML</a> is dense on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M330">View MathML</a>, passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M331">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M332">View MathML</a>

(24)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M333">View MathML</a>. If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M334">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M335">View MathML</a>, from (24) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M40">View MathML</a> in the generalized functions sense. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M337">View MathML</a>

holds and, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M338">View MathML</a>, it can be seen that (7) holds. In the above expressions, by C we denote generic constants which depend only on the given functions and Lebesgue measure of the domain D.

Now it remains to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M122">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M151">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M342">View MathML</a>, from (24) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M343">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M344">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M345">View MathML</a> in the generalized functions sense, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M346">View MathML</a>.

Moreover, from (19) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M347">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M348">View MathML</a> is the orthogonal projector. Since the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M349">View MathML</a> is orthogonal and complete in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M351">View MathML</a> strongly, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M352">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307">View MathML</a>. Then, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M304">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M68">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M358">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307">View MathML</a>. On the other hand, since the projection operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M348">View MathML</a> is self-adjoint in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M361">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M362">View MathML</a>

thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M363">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M307">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M365">View MathML</a>. The proof of Theorem 1 is complete. □

Proof of Lemma 1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M185">View MathML</a> is constant, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M367">View MathML</a>

to prove that condition (6) holds, we only need to show that g satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M368">View MathML</a>

By taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M369">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M189">View MathML</a>, for the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M371">View MathML</a>

given in (8) defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M373">View MathML</a>

and the proof is complete. □

Proof of Lemma 2 For

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M374">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M375">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M376">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199">View MathML</a>, for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M379">View MathML</a>

given in (12) defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M187">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M381">View MathML</a>

Hence, to prove that condition (6) holds, we only need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M382">View MathML</a>

holds. If we take into account again that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M383">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M199">View MathML</a>, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/202/mathml/M385">View MathML</a>

The proof is complete. □

Competing interests

The author declares that he has no competing interests.

References

  1. Isakov, V: Inverse Problems for Partial Differential Equations, Springer, New York (2006)

  2. Radon, J: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipz., Math.-Nat.wiss. Kl.. 69, 262–277 (1917)

  3. Natterer, F, Wübbeling, F: Mathematical Methods in Image Reconstruction, SIAM, Philadelphia (2001)

  4. Natterer, F: The Mathematics of Computerized Tomography, Vieweg+Teubner, Wiesbaden (1986)

  5. Cormack, AM: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys.. 34, 2722–2727 (1963). Publisher Full Text OpenURL

  6. Cormack, AM: The Radon transform on a family of curves in the plane. Proc. Am. Math. Soc.. 83, 325–330 (1981). Publisher Full Text OpenURL

  7. Natterer, F: Inversion of attenuated Radon transform. Inverse Probl.. 17, 113–119 (2001). Publisher Full Text OpenURL

  8. Novikov, RG: An inversion formula for the attenuated X-ray transformation. Ark. Mat.. 40(1), 145–167 (2002). Publisher Full Text OpenURL

  9. Quinto, ET: Radon transforms on curves in the plane. Tomography, Impedance Imaging and Integral Geometry, pp. 231–244. Am. Math. Soc., Providence (1994)

  10. Ambartsoumian, G, Gouia-Zarrad, R, Lewis, M: Inversion of the circular Radon transform on an annulus. Inverse Probl.. 26, Article ID 105015 (2010)

  11. Nguyen, MK, Truong, TT: Inversion of a new circular-arc Radon transform for Compton scattering tomography. Inverse Probl.. 26, Article ID 065005 (2010)

  12. Rigaud, G, Nguyen, MK, Louis, AK: Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography. Inverse Probl. Sci. Eng.. 20(6), 809–839 (2012). Publisher Full Text OpenURL

  13. Amirov, AK: Integral Geometry and Inverse Problems for Kinetic Equations, VSP, Utrecht (2001)

  14. Gelfand, IM, Gindikin, SG, Graev, MI: Selected Topics in Integral Geometry, Am. Math. Soc., Providence (2003)

  15. Romanov, VG: Integral Geometry and Inverse Problems for Hyperbolic Equations, Springer, Berlin (1974)

  16. Sharafutdinov, VA: Integral Geometry of Tensor Fields, VSP, Utrecht (1994)

  17. Arridge, SR: Optical tomography in medical imaging. Inverse Probl.. 15, R41–R93 (1999). Publisher Full Text OpenURL

  18. Bal, G: Inverse transport theory and applications. Inverse Probl.. 25, Article ID 053001 (2009)

  19. Stefanov, P: Inverse problems in transport theory. In: Uhlmann G (ed.) Inside Out: Inverse Problems and Applications, Cambridge University Press, Cambridge (2003)

  20. Tamasan, A: An inverse boundary value problem in two-dimensional transport. Inverse Probl.. 18, 209–219 (2002). Publisher Full Text OpenURL

  21. Anikonov, DS, Kovtanyuk, AE, Prokhorov, IV: Transport Equation and Tomography, VSP, Utrecht (2002)

  22. Case, KM, Zweifel, PF: Linear Transport Theory, Addison-Wesley, Reading (1967)

  23. Klibanov, MV, Yamamoto, M: Exact controllability for the time dependent transport equation. SIAM J. Control Optim.. 46(6), 2071–2095 (2007). Publisher Full Text OpenURL

  24. Amirov, AK: Existence and uniqueness theorems for the solution of an inverse problem for the transport equation. Sib. Math. J.. 27, 785–800 (1986)

  25. Amirov, A, Yildiz, M, Ustaoglu, Z: Solvability of a problem of integral geometry via an inverse problem for a transport-like equation and a numerical method. Inverse Probl.. 25, Article ID 095002 (2009)

  26. Ustaoglu, Z, Heydarov, B, Amirov, S: On the solvability and approximate solution of a two dimensional coefficient inverse problem for a transport-like equation. Inverse Probl.. 26, Article ID 115019 (2010)

  27. Amirov, A, Ustaoglu, Z, Heydarov, B: Solvability of a two dimensional coefficient inverse problem for transport equation and a numerical method. Transp. Theory Stat. Phys.. 40(1), 1–22 (2011). Publisher Full Text OpenURL

  28. Golgeleyen, I: An integral geometry problem along geodesics and a computational approach. An. Univ. “Ovidius” Constanţa, Ser. Mat.. 18(2), 91–112 (2010). PubMed Abstract OpenURL

  29. Golgeleyen, I: An inverse problem for a generalized transport equation in polar coordinates and numerical applications. Inverse Probl.. 29, Article ID 095006 (2013)

  30. Lavrent’ev, MM, Anikonov, YE: A certain class of problems in integral geometry. Sov. Math. Dokl.. 8, 1240–1241 (1967)

  31. Klibanov, MV, Timonov, A: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht (2004)

  32. Courant, R, Hilbert, D: Methods of Mathematical Physics. Partial Differential Equations, Interscience, New York (1962)

  33. Lavrent’ev, MM, Romanov, VG, Shishatskii, SP: Ill-Posed Problems of Mathematical Physics and Analysis, Am. Math. Soc., Providence (1986)

  34. Mikhailov, VP: Partial Differential Equations, Mir, Moscow (1978)

  35. Evans, LC: Partial Differential Equations, Am. Math. Soc., Providence (1998)

  36. Lions, JL, Magenes, E: Nonhomogeneous Boundary Value Problems and Applications, Springer, London (1972)