Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

Abstract

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron–Martin space is denoted by H. Let ν=e−Uμ, where U:X→R is a sufficiently regular convex and continuous function. In this paper we are interested in the W2,2 regularity of the weak solutions of elliptic equations of the typeλu−Lνu=f, where λ>0, f∈L2(X,ν) and Lν is the self-adjoint operator associated with the quadratic form(ψ,φ)↦∫X〈∇Hψ,∇Hφ〉Hdνψ,φ∈W1,2(X,ν).