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An eigenvalue optimization problem for the p-Laplacian

Part of: Elliptic equations and systems Variational methods for eigenvalues of operators

Published online by Cambridge University Press:  08 October 2015

Anisa M. H. Chorwadwala  and
Rajesh Mahadevan
Anisa M. H. Chorwadwala
Affiliation:
Indian Institute of Science Education and Research, Pune, India (anisa@iiserpune.ac.in)
Rajesh Mahadevan*
Affiliation:
Departamento de Matemática, Universidad de Concepción, Concepción, Chile (rmahadevan@udec.cl)
*
*Corresponding author.
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Abstract

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

Keywords

shape optimizationDirichlet p-Laplacianshape derivative analysismoving plane methodcomparison principles

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 49R05: Variational methods for eigenvalues of operators (should also be assigned at least one other classification number in Section 49)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 6 , December 2015 , pp. 1145 - 1151
Copyright
Copyright © Royal Society of Edinburgh 2015 

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