Counterexamples to maximal regularity for operators in divergence form

In this paper, we present counterexamples to maximal Lᵖ 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 L 2 -regularity on H⁻¹ 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 Lᵖ L p -regularity on H⁻¹(Rᵈ) H - 1 ( R d ) or L² L 2 -regularity on L²(Rᵈ) L 2 ( R d ) .