Extremal functions for a class of trace Trudinger-Moser inequalities

TL;DR

This paper establishes new extremal functions for trace Trudinger-Moser inequalities on compact Riemann surfaces with boundary, extending previous results through blow-up analysis techniques.

Contribution

It generalizes existing inequalities and finds extremals for a broader class of trace Trudinger-Moser inequalities on Riemann surfaces.

Findings

Established several new trace Trudinger-Moser inequalities.

Obtained extremal functions using blow-up analysis.

Extended previous results by Li-Liu and Yang.

Abstract

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].