Maximal regularity for gradient systems with boundary degeneracy

Abstract

We study a class of elliptic operators $L$ that degenerate at the boundary of a bounded open set $\mathcal{O}\subset {\mathbb{R}}^d$ and possess a symmetrizing invariant measure $\mu$. Such operators are associated with diffusion processes in $\mathcal{O}$ which are invariant for time reversal. After showing that the corresponding elliptic equation $\lambda \phi -L\phi =f$ has a unique weak solution for any $\lambda >0$ and $f\in L^2(\mathcal{O},\mu )$, we obtain new results for the characterization of the domain of $L$.