On non-autonomous maximal regularity for elliptic operators in divergence form

TL;DR

This paper establishes maximal regularity results for non-autonomous elliptic operators in divergence form under minimal fractional time-derivative conditions, extending previous results that required stronger assumptions.

Contribution

It proves maximal regularity in L2 for elliptic operators with coefficients having a fractional time-derivative of order one-half, improving upon prior requirements.

Findings

Maximal regularity achieved under fractional time-derivative bound

Applicable to Dirichlet, Neumann, and mixed boundary conditions

Extends previous results requiring higher-order derivatives

Abstract

We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset $\Omega$ $\subseteq$ R n. We obtain maximal regularity in L 2 ($\Omega$) if the coefficients are bounded, uniformly elliptic, and satisfy a scale invariant bound on their fractional time-derivative of order one-half. Previous results even for such forms required control on a time-derivative of order larger than one-half.