Asymptotic analysis for some linear eigenvalue problems via Gamma-Convergence

Abstract

This paper is devoted to the analysis of the asymptotic behaviour when the parameter $\lambda$ goes to $+\infty$ for operators of the form $-\Delta + \lambda a $ or, more generally, cooperative systems operators of the form $\left(\begin{smallmatrix} -\Delta+ {\lambda} a & -b \\ -c & -\Delta + {\lambda} d \end{smallmatrix} \right) $ where the potentials $a$ and $d$ vanish in some subregions of the domain $\Omega$. 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, as does the spectrum. The class of potentials considered here is fairly large, substantially improving previous results, allowing in particular ones that vanish on a Cantor set, 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.