Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains

Abstract

We consider an elliptic Kolmogorov equationu − Ku = f in a convex subset C of a separable Hilbert space X. The Kolmogorov operator K is a realization of u 7→ 1 Tr (D 2 u(x)) + hAx − DU(x),Du(x)i, A is a self-adjoint operator in X and U : X 7→R ∪ {+∞} is a convex function. We prove that for � > 0 and f ∈ L 2 (C,�) the weak solution u belongs to the Sobolev space W 2,2 (C,�), whereis the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of C. The general results are applied to Kolmogorov equations of reaction-diffusion and Cahn-Hilliard stochastic PDEs in convex sets of suitable Hilbert spaces.