The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition

Authors:
Antoine Henrot and Henrik Shahgholian

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2399-2416

MSC (1991):
Primary 35R35, 35J70, 76S05

DOI:
https://doi.org/10.1090/S0002-9947-02-02892-1

Published electronically:
February 14, 2002

MathSciNet review:
1885658

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Abstract | References | Similar Articles | Additional Information

Abstract: Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the $p$-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure $a(x)$ on the “free” streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function $a(x)$ is subject to certain convexity properties. In our earlier results we have considered the case of constant $a(x)$. In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the $p$-capacitary potentials in convex rings, with $C^1$ boundaries.

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Additional Information

**Antoine Henrot**

Affiliation:
Ecole des Mines and Institut Elie Cartan, UMR CNRS 7502 and INRIA BP 239, 54506 Vandoeuvre-les-Nancy Cedex, France

Email:
henrot@iecn.u-nancy.fr

**Henrik Shahgholian**

Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Email:
henriks@math.kth.se

Keywords:
Free boundary,
convexity,
non-linear joining conditions

Received by editor(s):
July 14, 2000

Received by editor(s) in revised form:
August 16, 2001

Published electronically:
February 14, 2002

Additional Notes:
The first author thanks Göran Gustafsson Foundation for several visiting appointments to RIT in Stockholm

The second author was partially supported by the Swedish Natural Science Research Council and STINT. He also thanks Institute Elie Cartan for their hospitality. Both authors thank A. Petrosyan for some crucial remarks

Article copyright:
© Copyright 2002
American Mathematical Society