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

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 − U μ and Ω = G − 1 ( − ∞ , 0 ] . In this paper we are interested in the W 2 , 2 regularity of the weak solutions of elliptic equations of the type (0.1) λ u − L ν , Ω u = f , where λ > 0 , f ∈ L 2 ( Ω , ν ) and L ν , Ω is the self-adjoint operator associated with the quadratic form ( ψ , φ ) ↦ ∫ Ω 〈 ∇ H ψ , ∇ H φ 〉 H d ν ψ , φ ∈ W 1 , 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 ( ∇ H u ) ( x ) , Tr ( ∇ H G ) ( x ) 〉 H = 0 , where ρ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and Tr is the trace operator. [ABSTRACT FROM AUTHOR]