An inverse problem for the equation $▵u=-cu-d$
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, p. 1181-1209

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

$▵u=-cu-d\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{1em}{0ex}}u=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega .$

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 $\left(c,d\right)$ 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 $\left(c,d\right)$ 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.

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

$▵u=-cu-d\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{1em}{0ex}}u=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega .$

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 $\left(c,d\right)$, 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 $\left(c,d\right)$, 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.

@article{AIF_1994__44_4_1181_0,
author = {Vogelius, Michael},
title = {An inverse problem for the equation $\triangle u=-cu-d$},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {44},
number = {4},
year = {1994},
pages = {1181-1209},
doi = {10.5802/aif.1429},
zbl = {0813.35136},
mrnumber = {95h:35246},
language = {en},
url = {http://www.numdam.org/item/AIF_1994__44_4_1181_0}
}

Vogelius, Michael. An inverse problem for the equation $\triangle u=-cu-d$. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1181-1209. doi : 10.5802/aif.1429. http://www.numdam.org/item/AIF_1994__44_4_1181_0/

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