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Symmetry breaking in problems involving semilinear equations

Pages: 219 - 228, Issue Special, September 2011

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Lucio Cadeddu - Dipartimento di Matematica e Informatica, Via Ospedale 72, 09124 Cagliari, Italy (email)
Giovanni Porru - Dipartimento di Matematica e Informatica, Via Ospedale 72, 09124 Cagliari, Italy (email)

Abstract: 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:  Rearrangements, Energy integral, Maximization, Symmetry breaking
Mathematics Subject Classification:  Primary: 49J20; Secondary: 35J25, 49K30

Received: July 2010;      Revised: March 2011;      Published: October 2011.