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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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
Proc. Amer. Math. Soc. 144 (2016), 3431-3440 Request permission

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.
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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