Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group

Abstract

Let $(\Sigma ,g)$ be a closed Riemannian surface, $W^{1,2}(\Sigma ,g)$ be the usual Sobolev space, $\text{G}$ be a finite isometric group acting on $(\Sigma ,g)$, and ${\mathcal{H}}_{\text{G}}$ be a function space including all functions $u\in W^{1,2}(\Sigma ,g)$ with ${\int }_{\Sigma }ud{v}_g=0$ and $u(\sigma (x))=u(x)$ for all $\sigma \in \text{G}$ and all $x\in \Sigma$. Denote the number of distinct points of the set $\{\sigma (x):\sigma \in \text{G}\}$ by $I(x)$ and $\ell ={\inf }_{x\in \Sigma }I(x)$. Let ${\lambda }_1^{\text{G}}$ be the first eigenvalue of the Laplace-Beltrami operator on the space ${\mathcal{H}}_{\text{G}}$. Using blow-up analysis, we prove that if $\alpha <{\lambda }_1^{\text{G}}$ and $\beta \le 4\pi \ell$, then there holds $$\underset{u\in {\mathcal{H}}_{\text{G}},{\int }_{\Sigma }|{\nabla }_{g}u{|}^{2}d{v}_g-\alpha {\int }_{\Sigma }{u}^{2}d{v}_g\le 1}{\sup }{\int }_{\Sigma }{e}^{\beta u^2}d{v}_g<\infty ;$$ if $\alpha <{\lambda }_1^{\text{G}}$ and $\beta >4\pi \ell$, or $\alpha \ge {\lambda }_1^{\text{G}}$ and $\beta >0$, then the above supremum is infinity; if $\alpha <{\lambda }_1^{\text{G}}$ and $\beta \le 4\pi \ell$, then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser \cite{Moser}, L. Fontana \cite{Fontana} and W. Chen \cite{Chen-90}.

24 pages