The sharp Faber-Krahn inequality
speaker: Guido De Philippis (SISSA, Trieste e New York University)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).
timetable:
Mon 7 Oct, 16:50 - 17:40, Aula Dini
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