Maximal regularity for non-autonomous equations with measurable dependence on time

2014/10/23 by Gallarati, Chiara, Veraar, Mark · 3 citations
#34G20 #35B65 #35K55 #35K90 #42B15 #42B20 #42B37 #47D06 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Secondary: 34G10

paper · doi:10.48550/arxiv.1410.6394

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∈ (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.

Cited by

• Trace theory for parabolic boundary value problems with rough boundary conditions
• H^∞-calculus for the Dirichlet Laplacian on conical domains
• Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains

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