Non-Autonomous Maximal Regularity for Forms of Bounded Variation

Abstract

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 $\mathcal{A}(t):V\to V^{\prime }$ is associated with a coercive, bounded, symmetric form $\mathfrak{a}(t,.,.):V\times V\to \mathbb{C}$ for all $t\in [0,T]$. Given $f\in L^2(0,T;H)$, $u_0\in V$ there exists always a unique solution $u\in MR(V,V^{\prime }):=L^2(0,T;V)\cap H^1(0,T;V^{\prime })$. 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 $g:[0,T]\to \mathbb{R}$ 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 $\mathcal{A}(t)$.

arXiv comment22 pages