Counterexamples to maximal regularity for operators in divergence form

Abstract

AbstractIn this paper, we present counterexamples to maximal $$L^p$$ 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 $$L^2$$ 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 $$L^p$$ L p -regularity on $$H^{-1}(\mathbb {R}^d)$$ H - 1 ( R d ) or $$L^2$$ L 2 -regularity on $$L^2(\mathbb {R}^d)$$ L 2 ( R d ) .