On maximal parabolic regularity for non-autonomous parabolic operators

TL;DR

This paper demonstrates that maximal parabolic regularity can be extended to non-autonomous parabolic problems with discontinuous time dependence, enabling broader applications in analysis and PDEs.

Contribution

It establishes the extrapolation of maximal parabolic regularity to a wider range of time integrability exponents for non-autonomous operators with discontinuous coefficients.

Findings

Maximal parabolic $L^r$-regularity holds for discontinuous non-autonomous second-order divergence form operators.

The results apply in very general geometric settings.

Existence results for related quasilinear equations are proved.

Abstract

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.