# The sharp Faber-Krahn inequality

## Guido De Philippis

### (ENS Lyon)

Date: Oct 07, 2013, time: 16:50

Abstract. The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume.
I will show a sharp quantitative enhancement of this result, confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman:
$\lambda_1(\Omega)-\lambda_1(B_1)\ge c_N \mathcal A (\Omega)^2\qquad \text{for all $\Omega\subset \mathbb R^N$ such that $|\Omega|=|B_1|$},$
where $\mathcal A(\Omega)$ is the Frankel asymmetry of a set:
$\mathcal A(\Omega)=\inf_{x_0\in \mathbb R^N} |\Omega \Delta B_1(x_0)|.$

More generally, the result applies to every optimal Poincar\'e-Sobolev constant for the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$. (Joint work with L. Brasco and B. Velichkov).

[admin]