Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary

In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface $ ( \Sigma,g) $ with smooth boundary $ \partial\Sigma $. To be exact, let $ \lambda_1(\partial\Sigma) $ denotes the first eigenvalue of the Laplace-Beltrami operator $ \Delta _ { g} $ on $ \partial \Sigma $. Moreover, for any $ 0\leq\alpha<\lambda_1(\partial\Sigma) $, we set $ \mathcal { H } = \{ u \in W^{1,2} ( \Sigma, g) : \left(\int _{\Sigma} |\nabla_g u|^2 dv_g -\alpha \int _{\partial\Sigma} {u^2}ds_g \right)^{1/2}\leq 1 \ \, \mathrm{and}\, \int _{\partial\Sigma} {u}\,ds_g = 0 \} $, where $ W^{1,2}(\Sigma, g) $ is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum

$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $

is attained by some function $ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [9] and Yang [19, 20].