An inverse problem for the equation $\triangle u=-cu-d$

Vogelius, Michael

Annales de l'Institut Fourier, Tome 44 (1994), p. 1181-1209 / Harvested from Numdam

Résumé

On considère un domaine plan convexe $\Omega $, dont la courbure du bord n’est pas trop dégénérée. Dans cet article, nous donnons des réponses partielles à une question d’identification liée au problème aux limites

▵u=-cu-dinΩ,u=0on∂Ω.

Nous montrons deux résultats : 1) Si $\Omega $ n’est pas une boule et si on considère seulement des solutions telles que $-cu-d\le 0$, alors il existe au plus un nombre dénombrable de paires de coefficients $(c,d)$, telles que la dérivée normale ${{\partial u} \over {\partial \nu }}\vert _{\partial \Omega }$ soit égale à une fonction $\psi \ne 0$ donnée.

2) Si aucune condition de signe n’est imposée, mais si on fait l’hypothèse supplémentaire que $\Omega $ est suffisamment différent d’une boule, alors, de nouveau, il existe au plus un nombre dénombrable de paires de coefficients $(c,d)$, telles que ${{\partial u} \over {\partial \nu }}\vert _{ \partial \Omega }$ soit égale à une fonction non-dégénérée $\psi $ donnée. Notre analyse est reliée à des travaux sur les conjectures de Pompeiu–Schiffer. Pour illustrer cette relation, nous montrons aussi comment notre analyse permet de montrer de manière très simple et rapide un résultat dû à Berenstein, concernant la conjecture de Schiffer.

Abstract

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

▵u=-cu-dinΩ,u=0on∂Ω.

We prove two results: 1) If $\Omega $ is not a ball and if one considers only solutions with $-cu-d\le 0$, then there exist at most finitely many pairs of coefficients $(c,d)$ so that the normal derivative ${{\partial u} \over {\partial \nu }}\vert _{ \partial \Omega }$ equals a given $\psi \ne 0$.

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 ${{\partial u} \over {\partial \nu }}\vert _{ \partial \Omega }$ 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.

Détail

Publié le : 1994-01-01

MR 95h:35246 | Zbl 0813.35136

DOI : https://doi.org/10.5802/aif.1429

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