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

Cuccu, Fabrizio; Emamizadeh, Behrouz; Porru, Giovanni

doi:10.1090/S0002-9939-08-09769-4

Optimization of the first eigenvalue in problems involving the $p$-Laplacian
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by Fabrizio Cuccu, Behrouz Emamizadeh and Giovanni Porru

Proc. Amer. Math. Soc. 137 (2009), 1677-1687

DOI: https://doi.org/10.1090/S0002-9939-08-09769-4

Published electronically: December 11, 2008

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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 $|\Omega |$, 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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