Dirichlet regularity and degenerate diffusion

Let $\Omega\subset\mathbb{R}^N$ be an open and bounded set and let $m\colon\Omega\rightarrow (0,\infty)$ be measurable and locally bounded. We study a natural realization of the operator $m \triangle$ in $C_0(\Omega):=\left\lbrace u\in C(\overline{\Omega}):\;u_{\vert\partial\Omega}=0\right\rbrace$. If $\Omega$ is Dirichlet regular, then the operator generates a positive contraction semigroup on $C_0(\Omega)$ whenever $\frac{1}{m}\in L^p_{\operatorname{loc}}(\Omega)$ for some $p>\frac{N}{2}$. If $m(x)$ does not go fast enough to 0 as $x\rightarrow\partial\Omega$, then Dirichlet regularity is necessary. However, if $\vert m(x)\vert\leq c\cdot \operatorname{dist}(x,\partial\Omega)^2$, then we show that $m \triangle_0$ generates a semigroup on $C_0(\Omega)$ without any regularity assumptions on $\Omega$. We show that the condition for degeneration of $m$ near the boundary is optimal.