Symmetry breaking and other features for Eigenvalue problems

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2011, 2011(Special): 61-70. doi: 10.3934/proc.2011.2011.61

Symmetry breaking and other features for Eigenvalue problems

Claudia Anedda ^1, and Giovanni Porru ^2,

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

Received January 2010 Revised February 2011 Published October 2011

Abstract
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Under a Creative Commons license

In the rst part of this paper we discuss a minimization problem where symmetry breaking arise. Consider the principal eigenvalue for the problem -$\Deltau = \lambdaxFu$ in the ball $B_(a+2) \subset\mathbb{R}^N$, where $N\>= 2$ and $F$ varies in the annulus $B_(a+2)\\B_a$, keeping a xed measure. We prove that, if a is large enough, the minimum of the corresponding principal eigenvalue is attained in a subset which is not symmetric. In the second part of the paper we show that the principal eigenvalue of an equation with an indenite weight in a general bounded domain $\Omega$ can be approximated by a functional related to the energy integral of a suitable nonlinear equation; furthermore, we show that a solution of this nonlinear equation approximates, in the $H^1(\Omega)$ norm, the principal eigenfunction of our problem.

Keywords: Minimization, Symmetry breaking, non- linear equations, Rearrangements, Eigenvalues.

Mathematics Subject Classification: Primary: 49K30; Secondary: 47A75; 35J25.

Citation: Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61

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