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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Optimization of the first eigenvalue in problems involving the $ p$-Laplacian


Authors: Fabrizio Cuccu, Behrouz Emamizadeh and Giovanni Porru
Journal: Proc. Amer. Math. Soc. 137 (2009), 1677-1687
MSC (2000): Primary 35P15, 47A75
Published electronically: December 11, 2008
MathSciNet review: 2470826
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Abstract: This paper concerns minimization and maximization of the first eigenvalue in problems involving the $ p$-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of $ N=2$ and $ p$ close to $ 2$, our equation models the vibration of a nonhomogeneous membrane $ \Omega$ which is fixed along the boundary. Given several materials (with different densities) of total extension $ \vert\Omega\vert$, we investigate the location of these material inside $ \Omega$ so as to minimize or maximize the first mode in the vibration of the membrane.


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Additional Information

Fabrizio Cuccu
Affiliation: Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Email: fcuccu@unica.it

Behrouz Emamizadeh
Affiliation: Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Email: porru@unica.it

Giovanni Porru
Affiliation: Department of Mathematics, The Petroleum Institute, P. O. Box 2533, Abu Dhabi, United Arab Emirates
Email: bemamizadeh@pi.ac.ae

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09769-4
PII: S 0002-9939(08)09769-4
Keywords: $p$-Laplacian, eigenvalues, shape optimization, rearrangements
Received by editor(s): June 28, 2007
Published electronically: December 11, 2008
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.