Maximal \(L^2\) regularity for Dirichlet problems in Hilbert spaces (Q387985)

The authors study the Dirichlet problem NEWLINE\[NEWLINE\lambda U- \mathcal{L}U=F \text{ in } \mathcal{O}, \qquad U=0 \text{ on } \partial \mathcal{O}. NEWLINE\]NEWLINE Here \( \mathcal{O}\) is an open subset of a Hilbert space \(X\), \(F\in L^2 (\mathcal{O},\mu)\), \(\mu\) is a non-degenerate centered Gaussian measure on \(X\), and \(\mathcal{L}\) is an Ornstein-Uhlenbeck operator.NEWLINENEWLINEThe main results of the paper concern the problem whether the weak solution \(U\) belongs to the Sobolev space \(W^{2,2} (\mathcal{O},\mu)\).NEWLINENEWLINEPrevious results show that this question has a positive answer in the case \(\mathcal{O}=X\), but on general domains \(\mathcal{O}\) an explicit basis of \( L^2 (\mathcal{O},\mu)\) formed by eigenfunctions is not available, that could play the role of the Hermite polynomials and Wiener chaos decomposition used in the case \(\mathcal{O}=X\). The authors follow a completely different approach that consists of two steps:NEWLINENEWLINEStep 1. Dimension-free \(W^{2,2}\) estimates for finite-dimensional approximations.NEWLINENEWLINEStep 2. Approximate weak solutions by cylindrical functions that solve the finite-dimensional problems.NEWLINENEWLINEBoth steps are rather delicate.NEWLINENEWLINEThe final result gives a sufficient condition in terms of geometric properties of the boundary \(\partial \mathcal{O}\). These conditions are defined via properties of a function that has \(\mathcal{O}\) as a level set. Examples discussed include half-spaces, regions below a graph, spheres, and the special case of \(X=L^2(0,1)\).