Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight. (14th August 2021)

• Record Type: Journal Article
• Authors: Anedda, Claudia; Cuccu, Fabrizio; Frassu, Silvia
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 4(2021)
• Page Start: 970
• Page End: 992
• Publication Date: 2021-08-14
• Subjects: 35R11 -- 47A75 -- 49R05; Fractional Laplacian -- eigenvalue problem -- optimization -- Steiner symmetry; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X20000267
• Languages: English
• ISSNs: 0008-414X

Abstract

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.