Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

Abstract

We consider the maximal regularity problem for non-autonomous evolution equations \begin{equation} \left\{ \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, τ] u(0)&=&u_0. \end{array} \right. \end{equation} Each operator $A(t)$ is associated with a sesquilinear form $\mathfrak{a}(t)$ on a Hilbert space $H$. We assume that these forms all have the same domain $V$. It is proved in \cite{HO14} that if the forms have some regularity with respect to $t$ (e.g., piecewise $\alpha$-Hölder continuous for some $\alpha >1/2$) then the above problem has maximal $L_p$--regularity for all $u_0$ in the real-interpolation space $(H,D(A(0)){)}_{1-1/p,p}$. In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference $\mathfrak{a}(t;\cdot ,\cdot )-\mathfrak{a}(s;\cdot ,\cdot )$ is continuous on a larger space than the common domain $V$. We give three examples which illustrate our results.

arXiv admin note: text overlap with arXiv:1402.1136