The second eigenvalue of the fractional $p-$Laplacian
| Autor: | Enea Parini, Lorenzo Brasco |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU) |
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Nonlocal eigenvalue problems
spectral optimization Disjoint sets Characterization (mathematics) [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] 01 natural sciences Omega NO Combinatorics symbols.namesake Mathematics - Analysis of PDEs FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] quasilinear nonlocal operators 0101 mathematics Caccioppoli estimates Analysis Applied Mathematics Eigenvalues and eigenvectors Mathematics Sequence 010102 general mathematics Mathematics::Spectral Theory 16. Peace & justice Functional Analysis (math.FA) 010101 applied mathematics Mathematics - Functional Analysis 35P30 47J10 35R09 Dirichlet boundary condition Bounded function symbols p-Laplacian Analysis of PDEs (math.AP) |
| Zdroj: | Advances in Calculus of Variation Advances in Calculus of Variation, Walter de Gruyter GmbH, 2016, ⟨10.1515/acv-2015-0007⟩ Advances in Calculus of Variation, 2016, ⟨10.1515/acv-2015-0007⟩ |
| ISSN: | 1864-8266 |
| DOI: | 10.1515/acv-2015-0007⟩ |
| Popis: | We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \[ \inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}. \] We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity. Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1 |
| Databáze: | OpenAIRE |
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