On an elliptic operator degenerating on the boundary

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$, let $D(x)\in C^\infty(\overline\Omega)$ be a defining function of the boundary, and let $B(x)\in C^\infty(\overline\Omega)$ be an $n\times n$ matrix function with self-adjoint positive definite values $B(x )=B^*(x)>0$ for all $x\in\overline\Omega$ The Friedrichs extension of the minimal operator given by the differential expression $\mathcal{A}_0=-\langle\nabla,D(x )B(x)\nabla\rangle$ to $C_0^\infty(\Omega)$ is described.