Global properties of generalized Ornstein-Uhlenbeck operators on Lp(RN, RN) with more than linearly growing coefficients

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:350

Article

Hieber, Matthias²  Lorenzi, Luca¹  Pruess, Jan³  Rhandi, Abdelaziz⁴  Schnaubelt, Roland⁵ 

[1] Univ Parma, Dept Matemat, I-43100 Parma, Italy

[2] Tech Univ Darmstadt, Fachbereich Math, D-64289 Darmstadt, Germany

[3] Univ Halle Wittenberg, FB Math & Informat, D-06099 Halle, Germany

[4] Univ Salerno, Dipartimento Ingn Informaz & Matemat Applicata, I-84084 Fisciano, SA, Italy

[5] Univ Karlsruhe, Fak Math, D-76128 Karlsruhe, Germany

关键词: Systems of elliptic PDEs;    Unbounded coefficients;    Strongly continuous semigroups;    L-p-L-q estimates;    Gradient L-p-estimates;   

DOI  :  10.1016/j.jmaa.2008.09.011

来源: Elsevier

【 摘 要 】

We show that the realization A(p) of the elliptic operator Au =div(Q del u)+ F (.) del u + Vu ill L-p (R-N. R-N), P is an element of vertical bar 1, +infinity vertical bar, generates a strongly continuous semigroup. and we determine its domain D(A(p)) = {u is an element of W-2,W-p(R-N.R-N): F (.) del u + del u is an element of L-p(R-N.R-N)} if 1 < p < +infinity. The diffusion coefficients Q = (q(ij)) are uniformly elliptic and bounded together with their first-order derivatives, the drift coefficients F can grow as vertical bar x vertical bar log vertical bar x vertical bar, and V can grow logarithmically. Our approach relies on the Monniaux-Pruss theorem on the sum of noncommuting operators. We also prove L-p-L-q estimates and, under somewhat stronger assumptions, we establish pointwise gradient estimates and smoothing of the semigroup in the spaces W-alpha.p (R-N, R-N), alpha is an element of vertical bar 0, 1 vertical bar, where 1 < p < +infinity. (c) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free