Counterexamples to maximal regularity for operators in divergence form

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 L²-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(R^d) or L²-regularity on L²(R^d).