Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 

 

Steiner symmetry in the minimization of the first eigenvalue in problems involving the $ p$-Laplacian


Authors: Claudia Anedda and Fabrizio Cuccu
Journal: Proc. Amer. Math. Soc. 144 (2016), 3431-3440
MSC (2010): Primary 35J20, 35P15, 47A75
DOI: https://doi.org/10.1090/proc/12972
Published electronically: February 2, 2016
MathSciNet review: 3503711
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega \subset \mathbb{R}^N$ be an open bounded connected set. We consider the eigenvalue problem $ -\Delta _p u =\lambda \rho \vert u\vert^{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 (\vert\Omega \vert-\gamma )$ for a fixed $ 0<\gamma <\vert\Omega \vert$. 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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J20, 35P15, 47A75

Retrieve articles in all journals with MSC (2010): 35J20, 35P15, 47A75


Additional Information

Claudia Anedda
Affiliation: Mathematics and Computer Science Department, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy
Email: canedda@unica.it

Fabrizio Cuccu
Affiliation: Mathematics and Computer Science Department, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy
Email: fcuccu@unica.it

DOI: https://doi.org/10.1090/proc/12972
Keywords: Eigenvalue problem, minimization, Steiner symmetry
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
Article copyright: © Copyright 2016 American Mathematical Society