# American Institute of Mathematical Sciences

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

## Symmetry breaking and other features for Eigenvalue problems

 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

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 inde nite 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.
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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