Geodesic
Some observations on the first eigenvalue of the $p$-Laplacian and its connections with asymmetry
Electronic Journal of Differential Equations, Tome 2001 (2001).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\lambda_1\ge\lambda_1^*(1+C\alpha(\Omega)^{3})$, 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.
Classification : 35J60, 35P30
Keywords: asymmetry, de giorgi perimeter, p-Laplacian, first eigenvalue, talenti's inequality 
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     author = {Bhattacharya, Tilak},
     title = {Some observations on the first eigenvalue of the $p${-Laplacian} and its connections with asymmetry},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2001},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a93/}
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Bhattacharya, Tilak. Some observations on the first eigenvalue of the $p$-Laplacian and its connections with asymmetry. Electronic Journal of Differential Equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a93/