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