A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

In this paper, on a compact Riemann surface $(\mathrm{\Sigma },g)$ with smooth boundary $\mathrm{\partial }\mathrm{\Sigma }$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let ${\lambda }_1(\mathrm{\Sigma })$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $$\mathcal{S}$= \left{ u \in W^{1,2} (\Sigma, g) : |\nabla_g u|2^2 \leq 1\right.$and$\left.\int\Sigma u ,dv_g = 0 \right },$where$W^{1,2}(\Sigma, g)$istheusualSobolevspace,$|\cdot|2$ denotes the standard $L^2$-norm and ${\mathrm{\nabla }}_g$ represent the gradient. By the method of blow-up analysis, we obtain \begin{eqnarray*} \sup{u \in $\mathcal{S}$} \int_{\Sigma} e^{ 2\pi u^{2} \left(1+\alpha|u|2^{2}\right) }d v{g} <+\infty, \ \forall \ 0 \leq\alpha<\lambda_1(\Sigma); \end{eqnarray*} when $\alpha \ge {\lambda }_1(\mathrm{\Sigma })$, the supremum is infinite. Moreover, we prove the supremum is attained by a function $u_{\alpha} \in C^\infty\left(\overline{\Sigma}\right)\cap \mathcal {S}$forsufficientlysmall$\alpha> 0$. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang \cite{Lu-Yang}, we strengthen the result of Yang \cite{Yang2006IJM}.