On a class of elliptic operators with unbounded coefficients in convex domains

Da Prato, Giuseppe ; Lunardi, Alessandra

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004), p. 315-326 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

We study the realization $A$ of the operator $\mathcal{A} =\frac{1}{2} \triangle - (DU, D\cdot)$ in $L^{2}(\Omega, \mu)$, where $\Omega$ is a possibly unbounded convex open set in $\mathbb{R}^{N}$, $U$ is a convex unbounded function such that $\lim_{x \rightarrow \partial \Omega, \, x \in \Omega} U(x) = + \infty$ and $\lim_{|x| \rightarrow + \infty, \, x \in \Omega} U(x) = + \infty$, $DU(x)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu(dx) = c \exp (-2 U(x)) dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D(A) = \{u \in H^{2}(\Omega,\mu): (DU, Du) \in L^{2}(\Omega,\mu)\}$ is a dissipative self-adjoint operator in $L^{2}(\Omega,\mu)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by $A$.

Détail

Publié le : 2004-12-01

MR 2148888 | Zbl 1162.35345

Bibliographie

[1] Bogachev, V.I. - Krylov, N.V. - Röckner, M., On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Part. Diff. Eqns., 26, 2001, 2037- 2080. | MR 1876411 | Zbl 0997.35012

[2] Brézis, H., Opérateurs maximaux monotones. North-Holland, Amsterdam1973.

[3] Cerrai, S., Second order PDE’s in finite and infinite dimensions. A probabilistic approach. Lecture Notes in Mathematics, 1762, Springer-Verlag, Berlin2001. | MR 1840644 | Zbl 0983.60004

[4] Davies, E.B., Heat kernels and spectral theory. Cambridge Univ. Press, Cambridge1989. | MR 990239 | Zbl 0699.35006

[5] Da Prato, G. - Lunardi, A., Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Diff. Eqns., 198, 2004, 35-52. | MR 2037749 | Zbl 1046.35025

[6] Da Prato, G. - Röckner, M., Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Relat. Fields, 124, 2002, 261-303. | MR 1936019 | Zbl 1036.47029

[7] Eberle, A., Uniqueness and non-uniqueness of singular diffusion operators. Lecture Notes in Mathematics, 1718, Springer-Verlag, Berlin1999. | MR 1734956 | Zbl 0957.60002

[8] Krylov, N.V., On Kolmogorov’s equations for finite-dimensional diffusions. In: G. Da Prato (ed.), Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Mathematics, 1715, Springer-Verlag, Berlin 1999, 1-64. | MR 1730228 | Zbl 0927.00037

[9] Lamberton, D., Equations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces $L^{p}$. J. Funct. Anal., 72, 1987, 252-262. | MR 886813 | Zbl 0621.47039

[10] Lunardi, A. - Vespri, V., Optimal $L^{\infty}$ and Schauder estimates for elliptic and parabolic operators with unbounded coefficients. In: G. Caristi - E. Mitidieri (eds.), Proceedings of the Conference Reaction-Diffusion Systems (Trieste 1995). Lect. Notes in Pure and Applied Math., 194, M. Dekker, New York 1998, 217-239. | MR 1472521 | Zbl 0887.47034

[11] Petersen, K., Ergodic Theory. Cambridge Univ. Press, Cambridge1983. | Zbl 0676.28008

[12] Röckner, M., $L^{p}$-analysis of finite and infinite dimensional diffusion operators. In: G. Da Prato (ed.), Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Mathematics, 1715, Springer-Verlag, Berlin 1999, 65-116. | Zbl 0944.60078

[13] Stannat, W., (Nonsymmetric) Dirichlet operators on $L^{1}$: existence, uniqueness and associated Markov processes. Ann. Sc. Norm. Sup. Pisa, Ser. IV, 28, 1999, 99-140. | MR 1679079 | Zbl 0946.31003

[14] Triebel, H., Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam1978. | MR 503903 | Zbl 0387.46032