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.