Non-Autonomous Maximal Regularity for Forms of Bounded Variation

We consider a non-autonomous evolutionary problem $$u' (t)+\mathcal A (t)u(t)=f(t), \quad u(0)=u_0,$$ where $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and the operator $\mathcal A (t)\colon V\to V^\prime$ is associated with a coercive, bounded, symmetric form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb{C}$ for all $t \in [0,T]$. Given $f \in L^2(0,T;H)$, $u_0\in V$ there exists always a unique solution $u \in MR(V,V'):= L^2(0,T;V) \cap H^1(0,T;V')$. The purpose of this article is to investigate when $u \in H^1(0,T;H)$. This property of maximal regularity in $H$ is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function $g \colon [0,T] \to \mathbb{R}$ such that \begin{equation*} \lvert\mathfrak{a}(t,u,v)- \mathfrak{a}(s,u,v)\rvert \le [g(t)-g(s)] \lVert u \rVert_V \lVert v \rVert_V \quad (s,t \in [0,T], s \le t). \end{equation*} In that case, we also show that $u(.)$ is continuous with values in $V$. Moreover we extend this result to certain perturbations of $\mathcal A (t)$.