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On the convexity and symmetry of solutions to an elliptic free boundary problem

Part of: Elliptic equations and systems Inequalities Partial differential equations

Published online by Cambridge University Press:  09 April 2009

Bernhard Kawohl
Bernhard Kawohl
Affiliation:
SFB 123, Universität Heidelberg, Im Neuenheimer Feld 294, D6900 Heidelberg, West Germany
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Abstract

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We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

MSC classification

Secondary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J20: Variational methods for second-order elliptic equations 35B50: Maximum principles 26D10: Inequalities involving derivatives and differential and integral operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 1 , February 1987 , pp. 57 - 68
Copyright
Copyright © Australian Mathematical Society 1987

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