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

2017; Elsevier BV; Volume: 458; Issue: 1 Linguagem: Inglês

10.1016/j.jmaa.2017.09.015

1096-0813

Gianluca Cappa, Simone Ferrari,

Advanced Mathematical Modeling in Engineering

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 typewhere λ > 0, f ∈ L 2 (Ω, ν) and L ν,Ω is the self-adjoint operator associated with the quadratic formIn 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)where ρ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure and Tr is the trace operator.