Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$, where $\Omega$ is knows as a bounded domain with smooth boundary $\partial D$. We choose any subdomain $D\subset Q$ with smooth boundary $\partial D$. Moreover, we suppose that there exists a subdomain $\omega$ of $\Omega$ and a subboundary $\Gamma$ of $\partial\Omega$ such that $$ \omega\times\{0\}\subset \partial D $$ and $$ \Gamma\times(\varepsilon,T-\varepsilon) \subset \partial D,\quad \varepsilon\in(0,T). $$
If we know three functions $g_0\in H^{\frac{3}{2},1}(\Gamma\times(\varepsilon,T-\varepsilon))$, $g_1\in H^{\frac{1}{2},1}(\Gamma\times(\varepsilon,T-\varepsilon))$ and $u_0\in L^2(\omega)$.
My question:
Can we extend the functions $g_0,g_1,u_0$ to a function $u\in H^{2,1}(D)$ such that $u|_{\omega\times\{0\}}=u_0$, $u|_{\Gamma\times(\varepsilon,T-\varepsilon)}=g_0$, $\partial_\nu u|_{\Gamma\times(\varepsilon,T-\varepsilon)}=g_1$ and we have the inequality $$ \|u\|_{H^{2,1}(D)} \le C\|g_0\|_{H^{\frac{3}{2},1}}+C\|g_1\|_{H^{\frac{1}{2},1}}+ C\|u_0\|_{L^2(\omega)}? $$ Here $\nu$ is the outer unit normal vector of $\partial\Omega$.
Thank you very much.
