  We consider the problem of maximal regularity for non-autonomous Cauchy
problems u ' (t) + A(t) u(t) = f (t), t $\in$ (0, $\tau$ ] u(0) = u 0. 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 H 1 2 with respect to the variable
t. This regularity assumption is optimal and our results are the most general
ones on this problem.

Analysis of PDEs;Functional Analysis