Calculus of Variations and Geometric Measure Theory

P. Álvarez-Caudevilla - A. Lemenant

Asymptotic analysis for some linear eigenvalue problems via Gamma-Convergence

created by lemenant on 26 Nov 2009
modified on 10 Feb 2015


Published Paper

Inserted: 26 nov 2009
Last Updated: 10 feb 2015

Journal: Adv. in Diff. Eq.
Year: 2010


This paper is devoted to the analysis of the asymptotic behaviour when the parameter $\lambda$ goes to infinity for operators of the form $-Delta + \lambda a $ or more generally, cooperative systems operators, where the potentials vanish in some subregions of the domain. We use the theory of Gamma-convergence, even for the non-variational cooperative system, to prove that for any reasonable bounded potentials $a$ and $d$ those operators converge in the strong resolvent sense to the operator in the vanishing regions of the potentials, so does the spectrum. The class of potentials considered here is fairly large substantially improving previous results, allowing in particular ones that vanish on Cantor sets, and forcing us to enlarge the class of domains to the so-called quasi-open sets. For the system various situations are considered applying our general result to the interplay of the vanishing regions of the potentials of both equations.