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Advances in Calculus of Variations

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Volume 9, Issue 4

Issues

The second eigenvalue of the fractional p-Laplacian

Lorenzo Brasco
  • Corresponding author
  • Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France
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/ Enea Parini
  • Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France
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Published Online: 2015-08-29 | DOI: https://doi.org/10.1515/acv-2015-0007

Abstract

We consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set Ωn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ2(Ω) is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem

inf{λ2(Ω):|Ω|=c}.

We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume c/2 whose mutual distance tends to infinity.

Keywords: Nonlocal eigenvalue problems; spectral optimization; quasilinear nonlocal operators; Caccioppoli estimates

MSC 2010: 35P30; 47J10; 35R09

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About the article


Received: 2015-02-20

Revised: 2015-07-05

Accepted: 2015-07-21

Published Online: 2015-08-29

Published in Print: 2016-10-01


Citation Information: Advances in Calculus of Variations, Volume 9, Issue 4, Pages 323–355, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0007.

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