Lions' maximal regularity problem with [formula omitted]-regularity in time

Abstract

We consider the problem of maximal regularity for non-autonomous Cauchy problems{u′(t)+A(t)u(t)=f(t),t∈(0,τ]u(0)=u0. The time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We are interested in J.L. Lions's problem concerning maximal regularity of such equations. We give a positive answer to this problem under minimal regularity assumptions on the forms. Our main assumption is that the forms are piecewise H12 with respect to the variable t. This regularity assumption is optimal and our results are the most general ones on this problem.