Lions' Maximal Regularity Problem at the Half-Holder Endpoint

Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic, real scalar coefficient, half-Holder in time and arbitrarily close to the heat equation, whose Lions solution has a time derivative that is not square integrable.

Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.