A trace Moser-Trudinger inequality on compact Riemannian surface with corners on its boundary

Abstract

In this paper, we establish a trace Moser-Trudinger inequality on a compact Riemann surface

\mathrm{\Sigma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma $$\end{document}

with corners on

\partial \mathrm{\Sigma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial \Sigma $$\end{document}

. It is also regarded as “trace imbedding inequality” when the boundary carries the conical sigularities. Besides, based on blow-up analysis, we prove the existence of extremal functions for this trace Moser-Trudinger inequality.

Keywords

Riemann surface with conical singularities / Blow-up analysis / Trace Moser-Trudinger inequality / Extremal function

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Funding

national natural science foundation of china(12001472)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature