Quantitative stability for the Trudinger-Moser inequality

TL;DR

This paper establishes quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane, linking the deficit to the distance from optimizers under specific conditions.

Contribution

It introduces new stability estimates for the Trudinger-Moser inequality, including a spectral gap and conditions under which the estimates hold, even in the critical case.

Findings

Quadratic control of the deficit by the distance to optimizers.

Stability estimates valid for small exponential growth or on round disks.

A new spectral gap result that may be of independent interest.

Abstract

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.