Maximal regularity for non-autonomous evolution equations

We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert space $H$ . We assume that these forms all have the same domain and satisfy some regularity assumption with respect to t (e.g., piecewise $\alpha$-H{\"o}lder continuous for some $\alpha\textgreater{} 1/2$). We prove maximal Lp-regularity for all initial values in the real-interpolation space $(H, D(A(0)))\_{1/p,p}$ . The particular case where $p = 2$ improves previously known results and gives a positive answer to a question of J.L. Lions [11] on the set of allowed initial data $u\_0$ .