Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

Chiara Gallarati, Mark Veraar

Research output: Contribution to journalArticleScientificpeer-review

16 Citations (Scopus)
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Abstract

In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an Lp(Lq)-theory for such equations for p,q∈(1,∞). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.
Original languageEnglish
Pages (from-to)1-41
Number of pages41
JournalPotential Analysis
DOIs
Publication statusPublished - 2016

Keywords

  • Singular integrals
  • Maximal Lp-regularity
  • Evolution equations
  • Functional calculus
  • Elliptic operators
  • Ap-weights
  • R-boundedness
  • Extrapolation
  • Quasi-linear PDE

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