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

Anedda, Claudia ; Cuccu, Fabrizio ; Frassu, Silvia

Canadian journal of mathematics, Tome 73 (2021) no. 4, pp. 970-992

Résumé

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.

Détail

DOI : 10.4153/S0008414X20000267

Mots-clés : Fractional Laplacian, eigenvalue problem, optimization, Steiner symmetry