Critical points of solutions to elliptic problems in planar domains

Jaime Arango; Adriana Gómez

doi:10.3934/cpaa.2011.10.327

Article Contents

2011, Volume 10, Issue 1: 327-338. Doi: 10.3934/cpaa.2011.10.327

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Critical points of solutions to elliptic problems in planar domains

Jaime Arango^1, and
Adriana Gómez^1,

1.
Departamento de Matemáticas, Universidad del Valle, Cali

Received: March 31, 2009

Revised: September 30, 2009

Published: December 2010

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Abstract

Given a planar domain $\Omega$, and an analytic function $f$, we describe the set of critical points for the solution $u$ of the semilinear elliptic problem $\Delta u = f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. For simply connected domains we establish that the set of critical points is finite while for non--simply connected domains we show that this set is made up of finitely many isolated points and finitely many analytic Jordan curves. Further results are given in the case that $\Omega$ is an annular domain whose border has nonzero curvature.

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Mathematics Subject Classification: Primary: 74K15; Secondary: 35J05, 65M06.

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