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

TL;DR

This paper improves the regularity conditions needed for maximal $L_p$-regularity in non-autonomous evolution equations governed by sesquilinear forms with less regularity in time.

Contribution

It shows that regularity assumptions can be weakened for certain sesquilinear forms, broadening the class of problems with maximal regularity.

Findings

Maximal $L_p$-regularity holds under weaker form regularity conditions.

The difference of forms is continuous on a larger space than the common domain.

Three examples illustrate the applicability of the new results.

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, \tau] 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 $\alpha$-H\"older 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.