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

Da Prato, Giuseppe; Lunardi, Alessandra

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1102-1123.

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

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

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