Geodesic
Some observations on the first eigenvalue of the \(p\)-Laplacian and its connections with asymmetry
Electronic journal of differential equations, Tome 2001 (2001)
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 
@article{EJDE_2001__2001__a15,
     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},
     year = {2001},
     volume = {2001},
     zbl = {0991.35032},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a15/}
}
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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__a15/