SOBOLEV REGULARITY FOR A CLASS OF SECOND ORDER ELLIPTIC PDE'S IN INFINITE DIMENSION

• 成果类型: Article
• 署名作者: Da Prato, Giuseppe; Lunardi, Alessandra
• 署名单位: Scuola Normale Superiore di Pisa; University of Parma
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP936
• 发表日期: 2014
• 页码: 2113-2160
• 关键词: invariant-measures; EQUATIONS; uniqueness; Operators; SPACES

摘要

We consider an elliptic Kolmogorov equation lambda u - Ku = f in a separable Hilbert space H. The Kolmogorov operator K is associated to an infinite dimensional convex gradient system: dX = (AX - DU (X)) dt + dW (t), where A is a self-adjoint operator in H, and U is a convex lower semicontinuous function. Under mild assumptions we prove that for lambda > 0 and f is an element of L-2(H, v) the weak solution u belongs to the Sobolev space W-2,W-2 (H, v), where v is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of u are proved. The maximal regularity results are used in the study of perturbed nongradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction diffusion and Cahn Hilliard stochastic PDEs.