A free boundary value problem in potential theory
Annales de l'Institut Fourier, Volume 25 (1975) no. 3-4, pp. 323-344.

This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain Ω and a function u defined in Ω satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of Ω. The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists a unique solution pair Ω, u. The second part of the paper is devoted to demonstrating that Ω is a smooth starshaped curve.

Le but de ce travail est de formuler et de résoudre un problème de frontière libre pour l’équation de Poisson à deux variables.

Le problème consiste à déterminer un domaine Ω et une fonction u définie dans Ω de façon que, dans Ω soit satisfaite l’équation et sur le bord Γ de Ω soient satisfaites en même temps une condition de Dirichlet et une condition du type de Neumann.

La méthode de résolution consiste à réduire ce problème à l’étude d’une inéquation variationnelle.

Avec des conditions convenables on montre qu’il y a une solution {Ω,u} unique; on démontre enfin que la courbe Γ est régulière et étoilée.

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     title = {A free boundary value problem in potential theory},
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Stampacchia, Guido; Kinderlehrer, D. A free boundary value problem in potential theory. Annales de l'Institut Fourier, Volume 25 (1975) no. 3-4, pp. 323-344. doi : 10.5802/aif.587. http://www.numdam.org/articles/10.5802/aif.587/

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