Faber–Krahn inequalities in sharp quantitative form

Abstract

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings W 1,2 0 (Ω) → L q (Ω).2010 Mathematics Subject Classification. 47A75, 49Q20, 49R05.

The quantitative isoperimetric inequality and related topics

“…The above conjecture on the optimal power in inequality (6.34) has been proved in a recent paper by Brasco et al [24]. Here is their result.…”

“…Though the approach of Cicalese and Leonardi to the quantitative isoperimetric inequality is based on the results of a difficult and deep theory, it has the advantage of providing a short proof that has been successfully applied to several other inequalities, see [1,18,19,24,47,48,74]. The proof we are going to present here is a further simplification of the original proof by Cicalese and Leonardi which has been developed in a more general context by Acerbi et al in [1].…”

“…With this proposition in hands it is now clear that the strategy followed in [24] is to prove a stability inequality for E( ) of the type (6.37). Namely they follow the pattern that we have discussed in Sect.…”

“…5.2 of proving first the stability estimate for nearly spherical sets and then to extend it to general open sets by a contradiction argument via regularity. The proof for nearly spherical sets is essentially a second variation argument and it leads to the following Fuglede type result, see [24,Th. 3.3].…”

Improved convergence theorems for bubble clusters. I. The planar case

“…Examples are the Euclidean isoperimetric inequality in higher codimension [BDF12], the isoperimetric inequalities on spheres and hyperbolic spaces [BDF12,BDF13], isoperimetric inequalities for eigenvalues [BDPV13], minimality inequalities of area minimizing hypersurfaces [DPM14b], and non-local isoperimetric inequalities [FFM + ]; moreover, in [FJ14] the same strategy is used to control by P (E) − P (B) a more precise distance from the family of balls (see also [Neu14] for the case of the Wulff inequality). are considered in the small volume regime m → 0 + .…”