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Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Published online by Cambridge University Press:  14 April 2020

Claudia Anedda
Affiliation:
Department of Mathematics and Computer Science, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy e-mail: canedda@unica.itsilvia.frassu@unica.it
Fabrizio Cuccu*
Affiliation:
Department of Mathematics and Computer Science, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy e-mail: canedda@unica.itsilvia.frassu@unica.it
Silvia Frassu
Affiliation:
Department of Mathematics and Computer Science, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy e-mail: canedda@unica.itsilvia.frassu@unica.it
*

Abstract

Let $\Omega \subset \mathbb {R}^N$ , $N\geq 2$ , be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta )^s u =\lambda \rho u$ in $\Omega $ with homogeneous Dirichlet boundary condition, where $(-\Delta )^s$ , $s\in (0,1)$ , is the fractional Laplacian operator, $\lambda \in \mathbb {R}$ and $ \rho \in L^\infty (\Omega )$ .

We study weak* continuity, convexity and Gâteaux differentiability of the map $\rho \mapsto 1/\lambda _1(\rho )$ , where $\lambda _1(\rho )$ is the first positive eigenvalue. Moreover, denoting by $\mathcal {G}(\rho _0)$ the class of rearrangements of $\rho _0$ , we prove the existence of a minimizer of $\lambda _1(\rho )$ when $\rho $ varies on $\mathcal {G}(\rho _0)$ . Finally, we show that, if $\Omega $ is Steiner symmetric, then every minimizer shares the same symmetry.

Type
Article
Copyright
© Canadian Mathematical Society 2020

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