Consider the solid cone $C$ as the region inside $z=1-\sqrt{x^2+y^2}$ and bounded by $0\leq z\leq 1$. Now let us define $$\Omega= \{z=\frac{1}{2}\} \cap C\quad \text{and}\quad \Sigma= (\partial C \cap \{z\leq \frac{1}{2}\}) \setminus \{z=0\}$$.

We want to solve the wave equation $$\Box u = \partial^2_z u-\partial^2_x u -\partial^2_y u =0$$ inside the region $$ C \cap \{z\leq \frac{1}{2}\}$$ subject to $$u|_{\Omega}=f, \quad \partial_z u|_{\Omega}=g,\quad u|_{\Sigma}=h.$$

Is this possible, say for any $(f,g,h) \in H^1(\Omega)\times L^2(\Omega)\times H^1(\Sigma)$ with the natural compatibility conditions for $f$ and $h$ on $\partial \Omega$?