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

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     title = {A free boundary value problem in potential theory},
     journal = {Annales de l'Institut Fourier},
     pages = {323--344},
     publisher = {Institut Fourier},
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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/

[1] C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica, Ann. di Mat. pura e appl., IV, 92 (1972), 107-127. | MR | Zbl

[2] V. Benci, On a filtration problem through a porous medium, Ann. di Mat. pura e appl., C (1974), 191-209. | MR | Zbl

[3] H. Brezis, Solutions with compact support of variational inequalities, Usp. Mat. Nauk, XXIX, 2 (176) (1974), 103-108. | MR | Zbl

[4] H. Brezis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana U. Math. J., 23,9 (1974), 831-844. | MR | Zbl

[5] H. Brezis and G. Stampacchia, Une nouvelle méthode pour l'étude d'écoulements stationnaires, CRAS, 276 (1973), 129-132. | MR | Zbl

[6] G. Duvaut, Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degré), CRAS, 276 (1973), 1461-1463. | MR | Zbl

[7] J. Frehse, On the regularity of the solution of a second order variational inequality, Boll. U.M.I., 6 (1972), 312-315. | MR | Zbl

[8] D. Kinderlehrer, The free boundary determined by the solution to a differential equation, to appear in Indiana Journal. | Zbl

[9] H. Lewy, On the nature of the boundary separating two domains with different regimes, to appear.

[10] H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, C.P.A.M., 22 (1969), 153-188. | MR | Zbl

[11] J.-L. Lions and G. Stampacchia, Variational Inequalities, C.P.A.M., 20 (1967), 493-519. | MR | Zbl

[12] G. Stampacchia, On the filtration of a fluid through a porous medium with variable cross section, Usp. Mat. Nauk., XXIX, 4 (178) (1974), 89-101. | MR | Zbl

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