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

Vogelius, Michael

Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 1181-1209.

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

$$▵u=-cu-d\text{in}\Omega ,u=0\text{on}\partial \Omega .$$

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 $\frac{\partial u}{\partial \nu }{|}_{\partial \Omega }$ 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 $\frac{\partial u}{\partial \nu }{|}_{\partial \Omega }$ 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.

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

$$▵u=-cu-d\text{in}\Omega ,u=0\text{on}\partial \Omega .$$

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

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