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Potential Analysis

, Volume 46, Issue 3, pp 527–567 | Cite as

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

  • Chiara Gallarati
  • Mark Veraar
Open Access
Article

Abstract

In this paper we study maximal L p -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 L p -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 L p (L q )-theory for such equations for \(p,q\in (1, \infty )\). 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.

Keywords

Singular integrals Maximal Lp-regularity Evolution equations Functional calculus Elliptic operators Ap-weights \(\mathcal {R}\)-boundedness Extrapolation Quasi-linear PDE 

Mathematics Subject Classification (2010)

Primary: 42B20 42B37; Secondary: 34G10 35B65 42B15 47D06 35K90 34G20 35K55 

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Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

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