A free boundary value problem in potential theory
Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, p. 323-344
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.
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.
@article{AIF_1975__25_3-4_323_0,
     author = {Stampacchia, Guido and Kinderlehrer, D.},
     title = {A free boundary value problem in potential theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {25},
     number = {3-4},
     year = {1975},
     pages = {323-344},
     doi = {10.5802/aif.587},
     zbl = {0303.31003},
     mrnumber = {58 \#22609},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1975__25_3-4_323_0}
}
Stampacchia, Guido; Kinderlehrer, D. A free boundary value problem in potential theory. Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, pp. 323-344. doi : 10.5802/aif.587. http://www.numdam.org/item/AIF_1975__25_3-4_323_0/

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