Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

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. Consider two sufficiently regular convex functions U:X→R and G:X→R. We let ν=e−Uμ and Ω=G−1(−∞,0]. In this paper we are interested in the W2,2 regularity of the weak solutions of elliptic equations of the type(0.1)λu−Lν,Ωu=f, where λ>0, f∈L2(Ω,ν) and Lν,Ω is the self-adjoint operator associated with the quadratic form(ψ,φ)↦∫Ω〈∇Hψ,∇Hφ〉Hdνψ,φ∈W1,2(Ω,ν). In addition we will show that if u is a weak solution of problem (0.1) then it satisfies a Neumann type condition at the boundary, namely for ρ-a.e. x∈G−1(0)〈Tr(∇Hu)(x),Tr(∇HG)(x)〉H=0, where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator.