A wave problem in a half-space with a unilateral constraint at the boundary

In this paper, we study the following problem: let $\Omega$ be a half-space of $\mathbb{R}^N$, defined by $\Omega = \{x = (x', x_N) \in\mathbb{R}^/x_N > \}$ where $x' = (x,\ldots, x_{N-1})$ is the usual notation, and let there be given functions $u_0\in H^1(\Omega)$ and $u_1 \in L^2(\Omega)$. We assume that $u_0|_{x_N=0}$ is nonnegative, and similarly $-(\partial u_0/\partial x_N)|_{x_N=0}$ (which is, a priori, an element of $H^{-1/2}(\mathbb{R}^{N-1})$) is nonnegative.