A sharp trace Adams' inequality in ${\mathbb{R}}^4$ and Existence of the extremals

Abstract

Let $\Omega \subseteq {\mathbb{R}}^4$ be a bounded domain with smooth boundary $\partial \Omega$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(\Omega )$ with zero mean value and zero Neumann boundary condition: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $\alpha \le 12{\pi }^2$. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space ${\mathbb{R}}_{+}^4$. With this classification result, we can show that $S(12{\pi }^2)$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary.

arXiv comment38 pages

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