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.

Mots clés : Kolmogorov operators in infinite dimension, maximal Sobolev regularity, Neumann boundary condition

[1] H. Airault and P. Malliavin. Intégration géométrique sur l’espace de Wiener. Bull. Sci. Math. 112 (1988) 3–52. | MR | Zbl

[2] L. Ambrosio, G. Savaré and L. Zambotti. Existence and stability for Fokker–Planck equations with log-concave reference measure. Probab. Theory Related Fields 145 (2009) 517–564. | DOI | MR | Zbl

[3] V. Barbu and G. Da Prato. The generator of the transition semigroup corresponding to a stochastic variational inequality. Comm. Partial Differential Equations 33 (2008) 1318–1338. | DOI | MR | Zbl

[4] V. Barbu, G. Da Prato and L. Tubaro. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space. Ann. Probab. 37 (2009) 1427–1458. | DOI | MR | Zbl

[5] V. Barbu, G. Da Prato and L. Tubaro. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 699–724. | DOI | Numdam | MR | Zbl

[6] V. I. Bogachev. Gaussian Measures. Amer. Math. Soc., Providence, 1998. | DOI | MR | Zbl

[7] H. Brézis. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973. | MR | Zbl

[8] P. Celada and A. Lunardi. Traces of Sobolev functions on regular surfaces in infinite dimensions. J. Funct. Anal. 266 (2014) 1948–1987. | DOI | MR | Zbl

[9] E. Cépa. Problème de Skorohod multivoque. Ann. Probab. 26 (1998) 500–532. | DOI | MR | Zbl

[10] G. Da Prato. An Introduction to Infinite Dimensional Analysis. Springer, Berlin, 2006. | DOI | MR | Zbl

[11] G. Da Prato, A. Debussche and L. Tubaro. Irregular semi-convex gradient systems perturbed by noise and application to the stochastic Cahn–Hilliard equation. Ann. Inst. Henri Poincaré Probab. Statist. 40 (2004) 73–88. | DOI | EuDML | Numdam | MR | Zbl

[12] G. Da Prato and A. Lunardi. Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differential Equations 198 (2004) 35–52. | DOI | MR | Zbl

[13] G. Da Prato and A. Lunardi. Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension. Ann. Probab. 42 (2014) 2113–2160. | DOI | MR | Zbl

[14] G. Da Prato and J. Zabczyk. Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Notes 293. Cambridge Univ. Press, Cambridge, 2002. | DOI | MR | Zbl

[15] A. Debussche and L. Zambotti. Conservative stochastic Cahn–Hilliard equation with reflection. Ann. Probab. 35 (2007) 1706–1739. | DOI | MR | Zbl

[16] J. Diestel and J. J. Uhl. Vector Measures. Mathematical Surveys 15. Amer. Math. Soc., Providence, RI, 1977. | DOI | MR | Zbl

[17] D. Feyel and A. De La Pradelle. Hausdorff measures on the Wiener space. Potential Anal. 1 (1992) 177–189. | DOI | MR | Zbl

[18] T. Funaki and S. Olla. Fluctuations for $\nabla \phi$ interface model on a wall. Stochastic Process. Appl. 94 (2001) 1–27. | DOI | MR | Zbl

[19] D. Nualart and E. Pardoux. White noise driven by quasilinear SPDE’s with reflection. Probab. Theory Related Fields 93 (1992) 77–89. | DOI | MR | Zbl

[20] M. Röckner, R.-C. Zhu and X.-C. Zhu. The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple. Ann. Probab. 40 (2012) 1759–1794. | DOI | MR | Zbl

[21] L. Zambotti. Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. Probab. Theory Related Fields 123 (2002) 579–600. | DOI | MR | Zbl