Counterexamples to maximal regularity for operators in divergence form

TL;DR

The paper investigates maximal $L^p$-regularity for parabolic equations in divergence form with time-space dependent coefficients, showing that maximal regularity on $H^{-1}$ or $L^2$ cannot in general be extended to $L^p$ or $H$-time derivatives for $p\neq 2$. Building on a construction due to Mooney, the authors produce counterexamples that exhibit failure of $L^r_tL^s_x$-integrability and, consequently, fail to yield $L^p(0,1;H^1)$-regularity or $u'\in L^2(0,1;H)$ in general. They present a negative result for Problem 1 (extrapolating maximal $L^2$-regularity to $p\neq 2$) and a negative result for Problem 2 (regularity of $u'$ in time) in the setting of elliptic operators in divergence form, even when the data are sufficiently regular. The findings underscore the sharpness of known positive results, emphasize the crucial role of coefficient regularity in time, and delineate the limitations of maximal regularity in non-autonomous divergence-form problems.

Abstract

In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$.

Theorems & Definitions (11)

• Theorem 2.1: Lions
• Theorem 2.2
• Remark 2.3
• Theorem 2.4: Failure of higher integrability
• proof
• Theorem 2.5: Failure of extrapolation of maximal $L^p$-regularity
• proof
• Remark 2.6
• Theorem 2.7: Failure of maximal $L^2$-regularity on $L^2({\mathbb R}^d)$
• proof