Counterexamples to maximal regularity for operators in divergence form

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 $L^2({\mathbb{R}}^d)$.

Published version including the referee's suggestions