Maximal $L^2$ regularity for Dirichlet problems in Hilbert spaces

TL;DR

This paper investigates the regularity of solutions to Dirichlet problems involving Ornstein-Uhlenbeck operators in infinite-dimensional Hilbert spaces, establishing conditions under which solutions have second-order Sobolev regularity.

Contribution

It provides new sufficient conditions based on boundary geometry for the regularity of solutions in infinite-dimensional settings, extending known finite-dimensional results.

Findings

Solutions belong to W^{2,2} when the domain is the whole space.

Regularity depends on boundary geometry in nontrivial ways.

For ball domains, regularity holds only for small radii.

Abstract

We consider the Dirichlet problem $\lambda U - {\mathcal{L}}U= F$ in \mathcal{O}, U=0 on $\partial \mathcal{O}$. Here $F\in L^2(\mathcal{O}, \mu)$ where $\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\mathcal{O}, \mu)$. It is well known that the question has positive answer if $\mathcal{O} = X$; if $\mathcal{O} \neq X$ we give a sufficient condition in terms of geometric properties of the boundary $\partial \mathcal{O}$. The results are quite different with respect to the finite dimensional case, for instance if \mathcal{O} is the ball centered at the origin with radius $r$ we prove that $U\in W^{2,2}(\mathcal{O}, \mu)$ only for small $r$.