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

TL;DR

This paper addresses Lions' maximal regularity problem for non-autonomous evolution equations, proving positive results under minimal piecewise H 1/2 regularity assumptions on the associated forms, thus establishing the most general conditions for such regularity.

Contribution

The paper provides the first positive resolution of Lions' problem under minimal piecewise H 1/2 regularity assumptions, establishing optimal regularity conditions for maximal regularity.

Findings

Maximal regularity holds under piecewise H 1/2 regularity of forms.

Results are the most general for this problem.

Optimal regularity assumptions are identified.

Abstract

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.