Symmetry breaking and other features for Eigenvalue problems

Pages: 61 - 70, Issue Special, September 2011

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Claudia Anedda - 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: 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.

Keywords:  Rearrangements, Eigenvalues, Minimization, Symmetry breaking, non- linear equations
Mathematics Subject Classification:  Primary: 49K30; Secondary: 47A75; 35J25

Received: January 2010;      Revised: February 2011;      Published: October 2011.