Nous considérons une équation de Kolmogorov elliptique dans un sous-ensemble convexe d’un espace de Hilbert séparable . L’opérateur de Kolmogorov est une réalisation de , où est un opérateur auto-adjoint dans et est une fonction convexe. Nous prouvons que pour et la solution faible appartient à l’espace de Sobolev , où est la mesure log-concave associée au système. Nous prouvons aussi des estimations maximales sur le gradient de qui permettent de montrer que satisfait des conditions au bord de Neumann au sens des traces à la frontière de . 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.
We consider an elliptic Kolmogorov equation in a convex subset of a separable Hilbert space . The Kolmogorov operator is a realization of , is a self-adjoint operator in and is a convex function. We prove that for and the weak solution belongs to the Sobolev space , where is the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of , that allow to show that satisfies the Neumann boundary condition in the sense of traces at the boundary of . The general results are applied to Kolmogorov equations of reaction–diffusion and Cahn–Hilliard stochastic PDEÕs in convex sets of suitable Hilbert spaces.
@article{AIHPB_2015__51_3_1102_0, author = {Da Prato, Giuseppe and Lunardi, Alessandra}, 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}, pages = {1102--1123}, publisher = {Gauthier-Villars}, volume = {51}, number = {3}, year = {2015}, doi = {10.1214/14-AIHP611}, mrnumber = {3365974}, zbl = {1330.35514}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP611/} }
TY - JOUR AU - Da Prato, Giuseppe AU - Lunardi, Alessandra TI - Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1102 EP - 1123 VL - 51 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP611/ DO - 10.1214/14-AIHP611 LA - en ID - AIHPB_2015__51_3_1102_0 ER -
%0 Journal Article %A Da Prato, Giuseppe %A Lunardi, Alessandra %T Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 1102-1123 %V 51 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP611/ %R 10.1214/14-AIHP611 %G en %F AIHPB_2015__51_3_1102_0
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, Tome 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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