Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

Julián Fernández Bonder; Analía Silva; Juan F. Spedaletti

doi:10.3934/dcds.2020355

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2021, Volume 41, Issue 5: 2125-2140. Doi: 10.3934/dcds.2020355

This issue Previous Article Minimal period solutions in asymptotically linear Hamiltonian system with symmetries Next Article Thermodynamic formalism of

$ \text{GL}_2(\mathbb{R}) $

-cocycles with canonical holonomies

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

1.
Instituto de Matemática Luis A. Santaló (IMAS), CONICET, Departamento de Matemática, FCEN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, C1428EGA, Av. Cantilo s/n, Buenos Aires, Argentina
2.
Instituto de Matemática Aplicada San luis (IMASL), Ejército de los Andes 950, D5700HHW, San Luis, Argentina

^* Corresponding author: J. Fernández Bonder
^* Corresponding author: J. Fernández Bonder

Received: December 31, 2019

Revised: August 31, 2020

Early access: October 2020

Published: May 2021

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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.

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Mathematics Subject Classification: Primary: 35P30, 35J92; Secondary: 49R05.

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