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

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

Gallarati, C. (TU Delft Analysis)
Veraar, M.C. (TU Delft Analysis)

2016

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.

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

http://resolver.tudelft.nl/uuid:ddb252d0-16cc-4ecd-8818-39c524c62ce2

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

0926-2601

Potential Analysis, 1-41

Institutional Repository

journal article

© 2016 C. Gallarati, M.C. Veraar