Symmetry breaking in problems involving semilinear equations

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2011, 2011(Special): 219-228. doi: 10.3934/proc.2011.2011.219

Symmetry breaking in problems involving semilinear equations

Lucio Cadeddu ^1, and Giovanni Porru ^1,

1.	Dipartimento di Matematica e Informatica, Via Ospedale 72, 09124 Cagliari, Italy

Received July 2010 Revised March 2011 Published October 2011

Abstract
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This paper is concerned with two maximization problems where symmetry breaking arises. The rst one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation -$\deltau = XF^u^q$ in the annulus $B_(a,a+2)$ of the plane. Here 0 q < 1 and F is a varying subset of $B_(a,a+2)$, with a fixed measure. We prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough. The second problem we consider is governed by the same equation in a disc $B_a+2$ when $F$ varies in the annulus Ba;a+2 keeping a xed measure. So, now we have a so called maximization problem with a constraint. As in the previous case, we prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough.

Keywords: Symmetry breaking, Maximization, Rearrangements, Energy integral.

Mathematics Subject Classification: Primary: 49J20; Secondary: 35J25, 49K30.

Citation: Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219

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