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

Matthias Georg Hieber, Luca Lorenzi, Jan Prüss, Abdelaziz Rhandi, Roland Schnaubelt

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14 Citations (Scopus)

Abstract

We show that the realization Ap of the elliptic operator A u = div (Q ∇ u) + F ṡ ∇ u + V u in Lp (RN, RN), p ∈ [1, + ∞ [, generates a strongly continuous semigroup, and we determine its domain D (Ap) = {u ∈ W2, p (RN, RN) : F ṡ ∇ u + V u ∈ Lp (RN, RN)} if 1 < p < + ∞. The diffusion coefficients Q = (qi j) are uniformly elliptic and bounded together with their first-order derivatives, the drift coefficients F can grow as | x | log | x |, and V can grow logarithmically. Our approach relies on the Monniaux-Prüss theorem on the sum of noncommuting operators. We also prove Lp-Lq estimates and, under somewhat stronger assumptions, we establish pointwise gradient estimates and smoothing of the semigroup in the spaces Wα, p (RN, RN), α ∈ [0, 1], where 1 < p < + ∞.

Original languageEnglish
Pages (from-to)100-121
Number of pages22
JournalJournal of Mathematical Analysis and Applications
Volume350
Issue number1
DOIs
Publication statusPublished - 2009 Feb 1
Externally publishedYes

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Keywords

  • Gradient L-estimates
  • L-L estimates
  • Strongly continuous semigroups
  • Systems of elliptic PDEs
  • Unbounded coefficients

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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