Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight
| Autor: | Claudia Anedda, Fabrizio Cuccu, Silvia Frassu |
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| Rok vydání: | 2020 |
| Předmět: |
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Pure mathematics General Mathematics Operator (physics) 010102 general mathematics 01 natural sciences Convexity 010101 applied mathematics symbols.namesake Bounded function Dirichlet boundary condition symbols Differentiable function 0101 mathematics Symmetry (geometry) Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Canadian Journal of Mathematics. 73:970-992 |
| ISSN: | 1496-4279 0008-414X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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