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 et une fonction définie dans de façon que, dans soit satisfaite l’équation et sur le bord de 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 unique; on démontre enfin que la courbe 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 and a function defined in satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of . 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 , . The second part of the paper is devoted to demonstrating that 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},
pages = {323--344},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {25},
number = {3-4},
year = {1975},
doi = {10.5802/aif.587},
zbl = {0303.31003},
mrnumber = {58 #22609},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.587/}
}
TY - JOUR AU - Stampacchia, Guido AU - Kinderlehrer, D. TI - A free boundary value problem in potential theory JO - Annales de l'Institut Fourier PY - 1975 SP - 323 EP - 344 VL - 25 IS - 3-4 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.587/ DO - 10.5802/aif.587 LA - en ID - AIF_1975__25_3-4_323_0 ER -
%0 Journal Article %A Stampacchia, Guido %A Kinderlehrer, D. %T A free boundary value problem in potential theory %J Annales de l'Institut Fourier %D 1975 %P 323-344 %V 25 %N 3-4 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.587/ %R 10.5802/aif.587 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.587/
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