Convexity of level curves for solutions to semilinear elliptic equations

David L. Finn

doi:10.3934/cpaa.2008.7.1335

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2008, Volume 7, Issue 6: 1335-1343. Doi: 10.3934/cpaa.2008.7.1335

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Convexity of level curves for solutions to semilinear elliptic equations

David L. Finn^1,

1.
Rose-Hulman Institute of Technology, Department of Mathematics, Terre Haute, IN 47803

Received: August 31, 2007

Revised: April 30, 2008

Published: October 2008

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Abstract

Let $\Omega$ be a bounded strictly convex planar domain, and $f$ be a smooth function satisfying $f(0) < 0$ and $f'(t) \geq 0$. In this paper, we provide a simple proof using just the maximum principle that the level curves of the unique positive solution to $\Delta u = f(u)$ in $\Omega$ satisfying $u = 0$ on $\partial\Omega$ are convex and there is a unique critical point. We also provide generalization of this result to cover certain cases with $f'(t) < 0$.

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Mathematics Subject Classification: Primary: 35B05; Secondary: 35J60, 35J65.

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