Inverse problems of recovering the boundary data with integral overdetermination conditions

In the present article we examine an inverse proble m of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various ta sks of mathematical physics: control of heat exchange prosesses and design of th ermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes o f reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equa tion of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regula r, i. e., all generalized derivatives occuring into the equation exists and a re summable to some power. The proof relies on the fixed point theorem and boo tstrap arguments. Stability estimates for solutions are also given. The solvabi lity conditions are close to necessary conditions.


Introduction
We consider the parabolic equation Inverse problems of recovering boundary regimes, in particular, the convective heat exchange problems are conventional (see, for instance, [2][3][4][5][6][7][8][9][10][11]). They arise in different problems of mathematical physics such as the problems of control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coverings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, the study of composite materials, etc. Mathematical models describing these prosesses and the corresponding inverse problems in both one-dimensional and multidimensional cases are described, for example, in [2]. The essential attention here is paid to numerical methods of solving inverse problems in question and some uniqueness theorems together stability estimates for solutions. We refer also to the monograph [3] mainly devoted to numerical methods of determining a solution, where in the one-dimensional case different inverse problems for parabolic equations and problems of recovering the boundary regimes as well are studied. The overdetrermination conditions are the values of a solution at some points lying inside the spatial domain. These problems are studied in different settings in dependence on the type of the ovedetermination conditions. It is often the case when these problems are ill-posed in the Hadamard sense. In particular, it is true in the case of the overdetermination conditions in the form of values of a solution at separate points or on some surfaces lying in the spatial domain (see [2]). At the present article we examine the problems with overdetermination conditions in the form of some integrals with weights of a solution over a spatial domain. Note that these conditions arise in applications and they are often used in literature. Inverse problems of recovering the right-hand side or coefficients of an equation with integral ovedetermination conditions are studied in the articles [12][13][14][15][16][17][18], the monographs [19,20], and some other articles. In particular, the existence and uniqueness theorem of a generalized solution to the problem (1)-(3) (from the class ) in the case of = 1 m and the Neumann boundary condition was obtained in [9] and a similar result for a heat-and-mass transfer system including the Navier-Stokes system and a parabolic equation for the concentration of an admixture was obtained in [10]. The article [11] is devoted to a regular solvability ( ) in the case of = 1 m and the Robin boundary conditions. The case of the Dirichlet boundary condition happens to be more complicated than the case of the Neumann (Robin) boundary conditions and was not studied before. The present article is devoted to this case. Under some conditions on the data we prove well-poseness of this problem. The article in some sense is an extension of [21], where the Robin boundary conditions are treated. Some our auxiliary statements are taken from this article.

Preliminaries
Let E be a Banach space. Denote by ( ; )  [21,23]). If = E R or = n E R then the latter space is . This class is a Banach space with the norm . We can define also the equivalent norm The equivalence results from Lemma 1 of the subsection 3.2.6 [1]. Similarly, we can define the spaces (0, ; ( )) The new norms after a possible change on a set of zero measure. If (0) = 0 q and q ɶ is an extension by zero of q for 0 t ≤ then where the constant 1 c is independent of (0, ] T τ ∈ and q . 2) The product q v ⋅ of functions in (0, ) Moreover, the following estimate holds: where the constant 3 c is independent of q but it depends on 0 δ and tends to ∞ as 0 0 δ → .
Suppose also that there exists a constant 0 > 0 δ such that The conditions on the data of the problem As a consequence of Proof. Extend the function g by zero for < 0 t and put ( , ), ( , )

Basic results
In addition to the above conditions we require that are linearly independent on Γ and 0 ( ) | u x Γ belongs to the span of these functions.

Theorem 3. Assume that G is a bounded domain with boundary of the class
By (16) (18), (19), and integrating by parts, we infer The last inequality can be written in either of the forms The function ω in (21) is a solution to the direct problem (18). The entries of B possess the property (0, ) and even more we have the inequality As was noticed in the proof of Lemma 1, the embedding theorems state that (0, ) Hence, we can assume that . In view of (15), we can write .
In view of our conditions on the coefficients, the last factor is estimated by some constant independent of δ . Estimate the first factor. We have The Newton-Leibnitz formula validates the inequality Estimate the second summand on the right-hand side of (25). To this end, we first make the change of variables 1 1 = t τ δ , 2 2 = t τ δ and next use the inequality In view of the condition (14) The vector-function a q satisfies the system (20), subtracting its k -th еquation from this equality and cancelling, we arrive at the equality = ', = 1, , ,   ω is a solution to the problem (34). A solution to the system vanishes for t δ ≤ . We obtain the same system with zero Cauchy data at the point = t δ and a new right-hand side F . Next, we repeat the same arguments and estimates on the segment [ , 2 ] δ δ .
Without loss of generality, we can assume that the constants arising in estimating the norm of the operator 0 R are the same. Thus, the system (35) is solvable on [ , 2 ] δ δ . Repeating the arguments on [2 ,3 ] δ δ and so on, we can construct a solution on the whole segment [0, ] T . The estimate in the claim of the theorem has been actually proven in the proof.
Remark. At first sight, the well-posedness conditions (15) look rather strange and possibly arising in the method of the proof. However, employing other methods leads actually to the same condtitions. It is possible that they are essential.