Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

TL;DR

This paper uses Gamma-convergence to analyze the asymptotic behavior of fractional eigenvalue problems, unifying various cases and extending known results for the fractional Laplacian as parameters vary.

Contribution

It introduces a unified Gamma-convergence framework to study the asymptotics of fractional eigenvalues, including new cases not previously analyzed.

Findings

Recovered known eigenvalue behaviors as fractional parameter s approaches 1.

Extended results for eigenvalues as p approaches infinity.

Analyzed new eigenvalue problems not previously covered.

Abstract

In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $p-$fractional laplacian when the fractional parameter $s$ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $p$ goes to $\infty$. Finally we analyze other eigenvalue problems not previously covered in the literature.