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

Journal article (2016)

Authors

C. Gallarati Analysis -

M.C. Veraar Analysis -

Research Group

Analysis () (TU Delft)

DOI: https://doi.org/10.1007/s11118-016-9593-7

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

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Published Date

2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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.

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