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November  2008, 7(6): 1335-1343. doi: 10.3934/cpaa.2008.7.1335

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, United States

Received  September 2007 Revised  May 2008 Published  August 2008

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 $f'(t) < 0$.
Keywords: Semilinear elliptic equation, convexity., maximum principle, level curves.
Mathematics Subject Classification: Primary: 35B05; Secondary: 35J60, 35J6.
Citation: David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
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