Maximal regularity for non-autonomous evolution equations

Abstract

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 $a(t;\ast ,\ast )$ 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{ö}lder continuous for some $α\textgreater{} 1/2$). 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 $u\_0$ .

19 pages. To appear in Math. Ann