A singular Moser-Trudinger inequality for mean value zero functions in dimension two

TL;DR

This paper establishes a new version of the Moser-Trudinger inequality for functions with mean value zero in two dimensions, incorporating a singular weight and extending previous results to a broader class of functions.

Contribution

It proves the finiteness and attainability of a singular Moser-Trudinger inequality for mean zero functions in two dimensions, generalizing prior work for the non-singular case.

Findings

Supremum of the weighted exponential integral is finite.

The supremum is attained by some function.

Extension of classical inequality to singular weights.

Abstract

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partial\Omega$. In this paper, we prove that for any $\beta\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(\Omega), \int_\Omega u dx=0, \int_\Omega|\nabla u|^2dx\leq1}\int_\Omega \frac{e^{2\pi(1-\beta) u^2}}{|x|^{2\beta}}dx$$ is finite and can be attained. This partially generalizes a well-known work of Alice Chang and Paul Yang (J. Differential Geom. 27 (1988), no. 2, 259-296) who have obtained the inequality when $\beta=0$.