Quantitative stability for the Trudinger-Moser inequality

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.

27 pages