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

Part of: Variational methods for eigenvalues of operators Miscellaneous topics - Partial differential equations General theory of linear operators

Published online by Cambridge University Press:  14 April 2020

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

Keywords

Fractional Laplacianeigenvalue problemoptimizationSteiner symmetry

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 47A75: Eigenvalue problems 49R05: Variational methods for eigenvalues of operators (should also be assigned at least one other classification number in Section 49)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 970 - 992
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Copyright
© Canadian Mathematical Society 2020

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