# A free boundary value problem in potential theory

Guido Stampacchia; D. Kinderlehrer

Annales de l'institut Fourier (1975)

- Volume: 25, Issue: 3-4, page 323-344
- ISSN: 0373-0956

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topStampacchia, Guido, and Kinderlehrer, D.. "A free boundary value problem in potential theory." Annales de l'institut Fourier 25.3-4 (1975): 323-344. <http://eudml.org/doc/74251>.

@article{Stampacchia1975,

abstract = {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 $\Omega $ and a function $u$ defined in $\Omega $ satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of $\Omega $. 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 $\Omega $, $u$. The second part of the paper is devoted to demonstrating that $\partial \Omega $ is a smooth starshaped curve.},

author = {Stampacchia, Guido, Kinderlehrer, D.},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3-4},

pages = {323-344},

publisher = {Association des Annales de l'Institut Fourier},

title = {A free boundary value problem in potential theory},

url = {http://eudml.org/doc/74251},

volume = {25},

year = {1975},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 25

IS - 3-4

SP - 323

EP - 344

AB - 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 $\Omega $ and a function $u$ defined in $\Omega $ satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of $\Omega $. 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 $\Omega $, $u$. The second part of the paper is devoted to demonstrating that $\partial \Omega $ is a smooth starshaped curve.

LA - eng

UR - http://eudml.org/doc/74251

ER -

## References

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- [2] V. BENCI, On a filtration problem through a porous medium, Ann. di Mat. pura e appl., C (1974), 191-209. Zbl0298.76049MR50 #9144
- [3] H. BREZIS, Solutions with compact support of variational inequalities, Usp. Mat. Nauk, XXIX, 2 (176) (1974), 103-108. Zbl0304.35036MR58 #1576
- [4] H. BREZIS and D. KINDERLEHRER, The smoothness of solutions to nonlinear variational inequalities, Indiana U. Math. J., 23,9 (1974), 831-844. Zbl0278.49011MR50 #13881
- [5] H. BREZIS and G. STAMPACCHIA, Une nouvelle méthode pour l'étude d'écoulements stationnaires, CRAS, 276 (1973), 129-132. Zbl0246.35021MR47 #4521
- [6] G. DUVAUT, Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degré), CRAS, 276 (1973), 1461-1463. Zbl0258.35037MR48 #6688
- [7] J. FREHSE, On the regularity of the solution of a second order variational inequality, Boll. U.M.I., 6 (1972), 312-315. Zbl0261.49021MR47 #7197
- [8] D. KINDERLEHRER, The free boundary determined by the solution to a differential equation, to appear in Indiana Journal. Zbl0336.35031
- [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. Zbl0167.11501MR40 #816
- [11] J.-L. LIONS and G. STAMPACCHIA, Variational Inequalities, C.P.A.M., 20 (1967), 493-519. Zbl0152.34601MR35 #7178
- [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. Zbl0312.76058MR54 #1853

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