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We consider the problem of maximal regularity for non-autonomous Cauchyproblems u ' (t) + A(t) u(t) = f (t), t $\in$ (0, $\tau$ ] u(0) = u 0. The timedependent operators A(t) are associated with (time dependent) sesquilinearforms on a Hilbert space H. We are interested in J.L. Lions's problemconcerning maximal regularity of such equations. We give a positive answer tothis problem under minimal regularity assumptions on the forms. Our mainassumption is that the forms are piecewise H 1 2 with respect to the variablet. This regularity assumption is optimal and our results are the most generalones on this problem.

Lions' maximal regularity problem with H 1/ 2 -regularity in time