Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula

TL;DR

This paper establishes sharp quantitative estimates for Faber-Krahn inequalities on various space forms, linking eigenvalue gaps to geometric and functional distances, and applies these results to enhance understanding of the Alt-Caffarelli-Friedman monotonicity formula.

Contribution

It provides the first sharp quantitative Faber-Krahn inequalities on the sphere and hyperbolic space, extending Euclidean results and introducing new spectral analysis techniques.

Findings

Eigenvalue gap controls geometric and eigenfunction distances.

New regularity results for perturbed Alt-Caffarelli functionals.

Quantitative bounds for the ACF monotonicity formula.

Abstract

The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. We prove that the gap between the first eigenvalue of a given set $\Omega$ and that of the ball quantitatively controls both the $L^1$ distance of this set from a ball {\it and} the $L^2$ distance between the corresponding eigenfunctions: \[ \lambda_1(\Omega) - \lambda_1(B) \gtrsim |\Omega \Delta B|^2 + \int |u_{\Omega} - u_B|^2, \] where $B$ denotes the nearest geodesic ball to $\Omega$ with $|B|=|\Omega|$ and $u_\Omega$ denotes the first eigenfunction with suitable normalization. On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon new regularity results for minimizers of critically perturbed Alt-Cafarelli type functionals in our companion paper. On the round sphere and hyperbolic space, the present results are the first sharp quantitative results with respect to any distance; here the local portion of the analysis is based on new implicit spectral analysis techniques. Second, we apply these sharp quantitative Faber-Krahn inequalities in order to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. We show that the energy drop in the ACF monotonicity formula from one scale to the next controls how close a pair of admissible functions is from a pair of complementary half-plane solutions. In particular, when the square root of the energy drop summed over all scales is small, our result implies the existence of tangents (unique blowups) of these functions.