Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains

Giuseppe Da Prato and Alessandra Lunardi

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Abstract

We consider an elliptic Kolmogorov equation $\lambda u-Ku=f$ in a convex subset $\mathcal{C}$ of a separable Hilbert space $X$. The Kolmogorov operator $K$ is a realization of $u\mapsto\frac{1}{2}\operatorname{Tr} [D^{2}u(x)]+\langle Ax-DU(x),Du(x)\rangle$, $A$ is a self-adjoint operator in $X$ and $U:X\mapsto\mathbb{R}\cup\{+\infty\}$ is a convex function. We prove that for $\lambda>0$ and $f\in L^{2}(\mathcal{C},\nu)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(\mathcal{C},\nu)$, where $\nu$ 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 $\mathcal{C}$. The general results are applied to Kolmogorov equations of reaction–diffusion and Cahn–Hilliard stochastic PDEÕs in convex sets of suitable Hilbert spaces.

Résumé

Nous considérons une équation de Kolmogorov elliptique $\lambda u-Ku=f$ dans un sous-ensemble convexe $\mathcal{C}$ d’un espace de Hilbert séparable $X$. L’opérateur de Kolmogorov $K$ est une réalisation de $u\mapsto\frac{1}{2}\operatorname{Tr} [D^{2}u(x)]+\langle Ax-DU(x),Du(x)\rangle$, où $A$ est un opérateur auto-adjoint dans $X$ et $U:X\mapsto\mathbb{R}\cup\{+\infty\}$ est une fonction convexe. Nous prouvons que pour $\lambda>0$ et $f\in L^{2}(\mathcal{C},\nu)$ la solution faible $u$ appartient à l’espace de Sobolev $W^{2,2}(\mathcal{C},\nu)$, où $\nu$ 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 $\mathcal{C}$. 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 1102-1123.

Dates
Received: 25 September 2013
Revised: 3 March 2014
Accepted: 3 March 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759241

Digital Object Identifier
doi:10.1214/14-AIHP611

Mathematical Reviews number (MathSciNet)
MR3365974

Zentralblatt MATH identifier
1330.35514

Subjects
Primary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 37L40: Invariant measures 35B65: Smoothness and regularity of solutions

Keywords
Kolmogorov operators in infinite dimension Maximal Sobolev regularity Neumann boundary condition

Citation

Da Prato, Giuseppe; Lunardi, Alessandra. Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 1102--1123. doi:10.1214/14-AIHP611. https://projecteuclid.org/euclid.aihp/1435759241


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