let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \partial^2_t u-\Delta u=0\,\quad &\text{on $(0,T)\times \Omega$}, \\ u=g\,\quad &\text{on $(0,T)\times \partial \Omega$,}\\ u(0,x)=\partial_t u(0,x)=0\quad &\text{on $\Omega$}\\ \end{cases} \end{aligned} \end{equation}
Is it true that the above equation admits a unique solution $$ u \in C^1(0,T;L^2(\Omega))\cap C(0,T;H^1(\Omega))$$ and that $\partial_\nu u \in L^{\infty}(0,T;H^{-\frac{1}{2}}(\partial \Omega))$, where $\nu$ is the normal unit vector field on $\partial \Omega$?
