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

On considère un domaine plan convexe Ω, 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 Ω n’est pas une boule et si on considère seulement des solutions telles que -cu-d≤0, alors il existe au plus un nombre dénombrable de paires de coefficients (c,d), telles que la dérivée normale ∂u ∂ν| ∂Ω soit égale à une fonction ψ≠0 donnée.

2) Si aucune condition de signe n’est imposée, mais si on fait l’hypothèse supplémentaire que Ω est suffisamment différent d’une boule, alors, de nouveau, il existe au plus un nombre dénombrable de paires de coefficients (c,d), telles que ∂u ∂ν| ∂Ω soit égale à une fonction non-dégénérée ψ 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 Ω 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 Ω is not a ball and if one considers only solutions with -cu-d≤0, then there exist at most finitely many pairs of coefficients (c,d) so that the normal derivative ∂u ∂ν| ∂Ω equals a given ψ≠0.

2) If one imposes no sign condition on the solutions but one additionally supposes that Ω is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients (c,d) so that ∂u ∂ν| ∂Ω equals a given non-degenerate ψ. 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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     author = {Vogelius, Michael},
     title = {An inverse problem for the equation $\triangle u=-cu-d$},
     journal = {Annales de l'Institut Fourier},
     pages = {1181--1209},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {4},
     year = {1994},
     doi = {10.5802/aif.1429},
     zbl = {0813.35136},
     mrnumber = {95h:35246},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1429/}
}
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Vogelius, Michael. An inverse problem for the equation $\triangle u=-cu-d$. Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 1181-1209. doi : 10.5802/aif.1429. https://aif.centre-mersenne.org/articles/10.5802/aif.1429/

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