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An inverse problem for the equation \(\Delta u=-cu-d\). (English) Zbl 0813.35136
Summary: Let \(\Omega\) be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem \[ \Delta u = -cu-d \;\text{in}\;\Omega,\quad u=0 \;\text{on}\;\partial \Omega. \] We prove two results:
1) If \(\Omega\) is not a ball and if one considers only solutions with \(-cu-d\leq 0\), then there exist at most finitely many pairs of coefficients \((c,d)\) so that the normal derivative \(\partial u/ \partial \nu|_{ \partial \Omega}\) equals a given \(\psi \neq 0\).
2) If one imposes no sign condition on the solutions but one additionally supposes that \(\Omega\) is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients \((c,d)\) so that \(\partial u/ \partial \nu|_{ \partial \Omega}\) equals a given non- degenerate \(\psi\).
Our analysis is related to work on the Pompeiu–Schiffer conjectures. To illustrate this relation we also show how our analysis provides a very elementary and short proof of a result, due to Berenstein, concerning the Schiffer conjecture.

MSC:
35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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