Advances in Differential Equations

Asymptotic analysis for some linear eigenvalue problems via Gamma-Convergence

Antoine Lemenant and Pablo Álvarez-Caudevilla

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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.

Article information

Adv. Differential Equations, Volume 15, Number 7/8 (2010), 649-688.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J10: Schrödinger operator [See also 35Pxx] 35J47: Second-order elliptic systems 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 49R05: Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)


Álvarez-Caudevilla, Pablo; Lemenant, Antoine. Asymptotic analysis for some linear eigenvalue problems via Gamma-Convergence. Adv. Differential Equations 15 (2010), no. 7/8, 649--688.

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