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

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 $$\underset{u\in W^{1,2}(\Omega ),{\int }_{\Omega }udx=0,{\int }_{\Omega }|\nabla u{|}^{2}dx\le 1}{\sup }{\int }_{\Omega }\frac{e^{2\pi (1-\beta )u^2}}{|x{|}^{2\beta }}dx$$ 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 $\beta =0$.

22 pages