Electron. J. Diff. Eqns., Vol. 2001(2001), No. 35, pp. 1-15.

Some observations on the first eigenvalue of the p-Laplacian and its connections with asymmetry

Tilak Bhattacharya

Abstract:
In this work, we present a lower bound for the first eigenvalue of the p-Laplacian on bounded domains in $\mathbb{R}^2$. Let $\lambda_1$ be the first eigenvalue and $\lambda_1^*$ be the first eigenvalue for the ball of the same volume. Then we show that , for some constant $C$, where $\alpha$ is the asymmetry of the domain $\Omega$. This provides a lower bound sharper than the bound in Faber-Krahn inequality.

Submitted September 3, 2000. Published May 16, 2001.
Math Subject Classifications: 35J60, 35P30.
Key Words: Asymmetry, De Giorgi perimeter, p-Laplacian, first eigenvalue, Talenti's inequality.

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Tilak Bhattacharya
Indian Statistical Institute
7, S.J.S. Sansanwal Marg
New Delhi 110 016 India
e-mail: tlk@isid.isid.ac.in

Current address:
Mathematics Department, Central Michigan University
Mount Pleasant, MI 48859 USA
e-mail: hadronT@netscape.net

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