Dirichlet regularity and degenerate diffusion

Abstract:

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.