Sobolev regularity for a class of second order elliptic PDE's in infinite dimension

TL;DR

This paper proves Sobolev regularity for solutions of a class of second order elliptic PDEs in infinite-dimensional spaces, with applications to stochastic PDEs like reaction-diffusion and Cahn-Hilliard equations.

Contribution

It establishes maximal Sobolev regularity for elliptic Kolmogorov equations in infinite dimensions under mild conditions, extending regularity theory to stochastic PDEs.

Findings

Solutions belong to Sobolev space W^{2,2}(H,ν) for λ>0 and f in L^2(H,ν)

Maximal gradient estimates are obtained for solutions

Results apply to stochastic PDEs such as reaction-diffusion and Cahn-Hilliard equations

Abstract

We consider an elliptic Kolmogorov equation $\lambda u - Ku = f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX = (AX - DU(X))dt + dW (t)$, where $A $ is a self--adjoint operator in $H$ and $U$ is a convex lower semicontinuous function. Under mild assumptions we prove that for $\lambda >0$ and $f\in L^2(H,\nu)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(H,\nu)$, where $\nu$ is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of $u$ are proved. The maximal regularity results are used in the study of perturbed non gradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction--diffusion and Cahn--Hilliard stochastic PDE's.