Steiner symmetry in the minimization of the first eigenvalue in problems involving the $p$-Laplacian
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- by Claudia Anedda and Fabrizio Cuccu PDF
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Abstract:
Let $\Omega \subset \mathbb {R}^N$ be an open bounded connected set. We consider the eigenvalue problem $-\Delta _p u =\lambda \rho |u|^{p-2}u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Delta _p$ is the $p$-Laplacian operator and $\rho$ is an arbitrary function that takes only two given values $0<\alpha <\beta$ and that is subject to the constraint $\int _\Omega \rho dx=\alpha \gamma +\beta (|\Omega |-\gamma )$ for a fixed $0<\gamma <|\Omega |$. The optimization of the map $\rho \mapsto \lambda _1(\rho )$, where $\lambda _1$ is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain $\Omega$ and we show that the minimizers inherit the same symmetry.References
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Additional Information
- Claudia Anedda
- Affiliation: Mathematics and Computer Science Department, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy
- MR Author ID: 740050
- Email: canedda@unica.it
- Fabrizio Cuccu
- Affiliation: Mathematics and Computer Science Department, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy
- MR Author ID: 689288
- Email: fcuccu@unica.it
- Received by editor(s): September 7, 2015
- Received by editor(s) in revised form: September 30, 2015
- Published electronically: February 2, 2016
- Communicated by: Joachim Krieger
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3431-3440
- MSC (2010): Primary 35J20, 35P15, 47A75
- DOI: https://doi.org/10.1090/proc/12972
- MathSciNet review: 3503711