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

Abstract

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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.

How to cite

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Stampacchia, 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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  1. [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. Zbl0258.76069MR53 #12207
  2. [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. [3] H. BREZIS, Solutions with compact support of variational inequalities, Usp. Mat. Nauk, XXIX, 2 (176) (1974), 103-108. Zbl0304.35036MR58 #1576
  4. [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. [5] H. BREZIS and G. STAMPACCHIA, Une nouvelle méthode pour l'étude d'écoulements stationnaires, CRAS, 276 (1973), 129-132. Zbl0246.35021MR47 #4521
  6. [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. [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. [8] D. KINDERLEHRER, The free boundary determined by the solution to a differential equation, to appear in Indiana Journal. Zbl0336.35031
  9. [9] H. LEWY, On the nature of the boundary separating two domains with different regimes, to appear. 
  10. [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. [11] J.-L. LIONS and G. STAMPACCHIA, Variational Inequalities, C.P.A.M., 20 (1967), 493-519. Zbl0152.34601MR35 #7178
  12. [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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