Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 3, pp. 1102-1123.

We consider an elliptic Kolmogorov equation λu-Ku=f in a convex subset 𝒞 of a separable Hilbert space X. The Kolmogorov operator K is a realization of u1 2Tr[D 2 u(x)]+Ax-DU(x),Du(x), A is a self-adjoint operator in X and U:X{+} is a convex function. We prove that for λ>0 and fL 2 (𝒞,ν) the weak solution u belongs to the Sobolev space W 2,2 (𝒞,ν), where ν is the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of 𝒞. The general results are applied to Kolmogorov equations of reaction–diffusion and Cahn–Hilliard stochastic PDEÕs in convex sets of suitable Hilbert spaces.

Nous considérons une équation de Kolmogorov elliptique λu-Ku=f dans un sous-ensemble convexe 𝒞 d’un espace de Hilbert séparable X. L’opérateur de Kolmogorov K est une réalisation de u1 2Tr[D 2 u(x)]+Ax-DU(x),Du(x), où A est un opérateur auto-adjoint dans X et U:X{+} est une fonction convexe. Nous prouvons que pour λ>0 et fL 2 (𝒞,ν) la solution faible u appartient à l’espace de Sobolev W 2,2 (𝒞,ν), où ν est la mesure log-concave associée au système. Nous prouvons aussi des estimations maximales sur le gradient de u qui permettent de montrer que u satisfait des conditions au bord de Neumann au sens des traces à la frontière de 𝒞. Les résultats généraux sont appliqués aux équations de réaction–diffusion de Kolmogorov et à l’équation de Cahn–Hilliard stochastique dans des ensembles convexes d’espaces de Hilbert appropriés.

DOI: 10.1214/14-AIHP611
Keywords: Kolmogorov operators in infinite dimension, maximal Sobolev regularity, Neumann boundary condition
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     title = {Maximal {Sobolev} regularity in {Neumann} problems for gradient systems in infinite dimensional domains},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Da Prato, Giuseppe; Lunardi, Alessandra. Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 3, pp. 1102-1123. doi : 10.1214/14-AIHP611. http://www.numdam.org/articles/10.1214/14-AIHP611/

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