Critical Trace Trudinger-Moser Inequalities on a Compact Riemann Surface with Smooth Boundary

Abstract

In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ∂Σ. Explicitly, let ${\lambda _1}\left( {\partial \sum } \right) = \mathop {\inf }\limits_{u \in {W^{1,2}}\left( {\sum ,g} \right),\int_{\partial \sum } {u{\rm{d}}{s_g} = 0,u\not \equiv 0} } {{\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} + {u^2}} \right){\rm{d}}{v_g}} } \over {\int_{\partial \sum } {{u^2}{\rm{d}}{s_g}} }}$ and ${\cal H} = \left\{ {u \in {W^{1,2}}\left( {\sum ,g} \right):\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} + {u^2}} \right){\rm{d}}{v_g} - \alpha \int_{\partial \sum } {{u^2}{\rm{d}}{s_g} \le 1\,\,\,\,\,{\rm{and}}\,\,\,\int_{\partial \sum } {u\,{\rm{d}}{s_g} = 0} } } } \right\},$ where W ^1,2(Σ, g) denotes the usual Sobolev space and ∇ _g stands for the gradient operator. By the method of blow-up analysis, we obtain $\mathop {\sup }\limits_{u \in {\cal H}} \int_{\partial \sum } {{{\rm{e}}^{\pi {u^2}}}{\rm{d}}{s_g}} \left\{ {\matrix{{ < + \infty ,} \hfill & {0 \le \alpha < {\lambda _1}\left( {\partial \sum } \right),} \hfill \cr { = + \infty ,} \hfill & {\alpha \ge {\lambda _1}\left( {\partial \sum } \right).} \hfill \cr } } \right.$ Moreover, the author proves the above supremum is attained by a function ${u_\alpha } \in {\cal H} \cap \,{C^\infty }\left( {\overline \sum } \right)$ for any 0 ≤ α < λ₁(∂Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang, Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary, Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015, 3161–3193].

Keywords

Trudinger-Moser inequality / Riemann surface / Blow-up analysis / Extremal function

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