Extremal functions for a class of trace Trudinger-Moser inequalities

In this paper, we concern trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. This kind of inequalities were extensively studied by Osgood-Phillips-Sarnak [24], Liu [20], Li-Liu [17], Yang [31, 32] and others. We establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals via the method of blow-up analysis. The results in the current paper generalize those of Li-Liu [17] and Yang [32, 33].