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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
    Journal of Mathematical Analysis and Applications, 2018
    Co-Authors: Gianluca Cappa, S Ferrari
    Abstract:

    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 − 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.

  • maximal sobolev regularity for solutions of elliptic equations in banach spaces endowed with a weighted Gaussian Measure the convex subset case
    arXiv: Analysis of PDEs, 2016
    Co-Authors: Gianluca Cappa, S Ferrari
    Abstract:

    Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian Measure $\mu$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} \lambda u-L_{\nu,\Omega} u=f, \end{align} where $\lambda>0$, $f\in L^2(\Omega,\nu)$ and $L_{\nu,\Omega}$ is the self-adjoint operator associated with the quadratic form \[(\psi,\phi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\phi\rangle_Hd\nu\qquad\psi,\phi\in W^{1,2}(\Omega,\nu).\] In addition we will show that if $u$ is a weak solution of problem $\lambda u-L_{\nu,\Omega} u=f$, with $\lambda>0$ and $f\in L^2(\Omega,\nu)$, then it satisfies a Neumann type condition at the boundary, namely for $\rho$-a.e. $x\in G^{-1}(0)$ \[\left\langle\,\text{Tr}\,(\nabla_Hu)(x),\,\text{Tr}\,(\nabla_H G)(x)\right\rangle_H=0,\] where $\rho$ is the Feyel--de La Pradelle Hausdorff--Gauss surface Measure and $\text{Tr}$ is the trace operator.