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

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({\alpha})=\underset{\int_{\Omega}udx=0,\frac{\partial u}{\partial\nu}|_{\partial\Omega}=0,\Vert\Delta u\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(\Omega)\setminus\{0\}}}{\sup}}\int_{\partial \Omega} e^{\alpha u^{2}}d\sigma<\infty \end{equation*} holds if and only if $\alpha\leq12\pi^2$. 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.