Recovery of the Boundary Data from a Nonlocal Dirichlet Boundary Condition Accompanied of a Neumann-Type Shape Condition for a Linear 2nd Order Elliptic Problem
by
Marian Slodicka
Institute of Applied Mathematics, Faculty of Mathematics and Physics, Comenius University, Bratislava, Slovak Republic
Coauthors: Hennie De Schepper (Department of Mathematics, Faculty of Engineering, University of Gent, Galglaan 2, B-9000 Ghent, Belgium)

Problems of parameter identification from nonstandard boundary conditions (BCs) in boundary value problems (BVPs), originating from various engineering disciplines, are of growing interest. Standard BCs which are prescribed pointwise are not always appropriate, as in some physical contexts only the average value of the solution u or of the total flux q ·n can be measured along some boundary part \Gamma. In the case that \int_\Gamma u or \int_\Gamma q ·n is prescribed, we speak of a nonlocal BC. In order to determine the solution uniquely, this type of BC must be accompanied of some additional information, e.g., about the shape of u or of q·n along \Gamma, resulting from other physical arguments.

In our paper we consider the following type of nonlocal BC

q·n = q (given) +c (unknown) ó õ \Gamma  u = U (given) ü ï ý ï þ on \Gamma,

where q represents a given shape function and c is an unknown real constant, which has to be determined as a part of the problem. Such a BC appears in the modelling of some electromagnetic field problems in electric machinery. Moreover, it also has a physical interpretation in the context of soil venting. Considering a single well experiment in a homogeneous soil matrix, one can assume that the inflow of air into the well tube will be constant, but unknown, while the average pressure at the well is given.

The main goal of this paper is to study a linear elliptic PDE of second order with BCs including the above type. We prove the existence and the uniqueness of the solution, and we outline a method for constructing it in terms of the solution of some auxiliary BVPs with standard (i.e. pointwise) BCs. In particular, the unknown flux and the trace of the solution at \Gamma are determined.

Date received: July 4, 1999