---
abstract: |
    We consider evolution equations of the form $$\label{Abstract equation}
    \dot u(t)+{\mathcal{A}}(t)u(t)=0,\ \ t\in[0,T],\ \
    u(0)=u_0,$$ where ${\mathcal{A}}(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form ${\mathfrak{a}}(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces $V, H$ and on the dual space $V'$ of $V.$ The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.
address:
- 'University of Wuppertal School of Mathematics and Natural Sciences Arbeitsgruppe Funktionalanalysis, Gaußstraße 20 D-42119 Wuppertal, Germany'
- ' Ibn Zohr University, Faculty of Sciences Departement of mathematics, Agadir, Morocco'
author:
- 'Omar EL-Mennaoui and Hafida Laasri\'
title: 'On the norm-continuity for evolution family arising from non-autonomous forms$^*$[^1]'
---

Introduction\[s1\] {#introductions1 .unnumbered}
==================

Throughout this paper $H,V$ are two separable Hilbert spaces over $\mathbb C$ such that $V$ is densely and continuously embedded into $H$ (we write $V \underset d \hookrightarrow H$). We denote by $(\cdot {\, \vert \,}\cdot)_V$ the scalar product and $\|\cdot\|_V$ the norm on $V$ and by $(\cdot {\, \vert \,}\cdot)_H, \|\cdot\|_H$ the corresponding quantities in $H.$ Let $V'$ be the antidual of $V$ and denote by $\langle ., . \rangle$ the duality between $V'$ and $V.$ As usual, by identifying $H$ with $H'$ we have $V\underset d\hookrightarrow H\cong H'\underset d\hookrightarrow V'$ see e.g., [@Bre11].

Let ${\mathfrak{a}}:[0,T]\times V \times V \to {\mathbb{C}}$ be a *non-autonomous sesquilinear form*, i.e., ${\mathfrak{a}}(t;\cdot,\cdot)$ is for each $t\in[0,T]$ a sesquilinear form, $$\label{measurability} 
{\mathfrak{a}}(\cdot;u,v) \text{ is measurable for all } u,v\in V,$$ such that $$\label{eq:continuity-nonaut}
|{\mathfrak{a}}(t,u,v)| \le M \Vert u\Vert_V \Vert v\Vert_V\ \text{ and } {\operatorname{Re}}~{\mathfrak{a}}(t,u,u)\ge \alpha \|u\|^2_V  \quad \ (t,s\in[0,T], u,v\in V),$$ for some constants $M, \alpha>0$ that are independent of $t, u,v.$ Under these assumptions there exists for each $t\in[0,T]$ an isomorphism ${\mathcal{A}}(t):V\to V^\prime$ such that $\langle {\mathcal{A}}(t) u, v \rangle = {\mathfrak{a}}(t,u,v)$ for all $u,v \in V.$ It is well known that $-{\mathcal{A}}(t),$ seen as unbounded operator with domain $V,$ generates an analytic $C_0$-semigroup on $V'$. The operator ${\mathcal{A}}(t)$ is usually called the operator associated with ${\mathfrak{a}}(t,\cdot,\cdot)$ on $V^\prime.$ Moreover, we associate an operator $A(t)$ with ${\mathfrak{a}}(t;\cdot,\cdot)$ on $H$ as follows $$\begin{aligned}
D(A(t))={}&\{u\in V {\, \vert \,}\exists f\in H \text{ such that } {\mathfrak{a}}(t;u,v)=(f{\, \vert \,}v)_H  \text{ for all } v\in V \}\\
A(t) u = {}&  f.\end{aligned}$$ It is not difficult to see that $A(t)$ is the part of ${\mathcal{A}}(t)$ in $H.$ In fact, we have $D(A(t))= \{ u\in V : {\mathcal{A}}(t) u \in H \}$ and $A(t) u =  {\mathcal{A}}(t) u.$ Furthermore, $-A(t)$ with domain $D(A(t))$ generates a holomorphic $C_0$-semigroup on $H$ which is the restriction to $H$ of that generated by $-{\mathcal{A}}(t).$ For all this results see e.g. [@Tan79 Chapter 2] or [@Ar06 Lecture 7].

We now assume that there exist $0< \gamma<1$ and a continuous function $\omega:[0,T]\longrightarrow [0,+\infty)$ such that $$\label{eq 1:Dini-condition} 
  |{\mathfrak{a}}(t,u,v)-{\mathfrak{a}}(s,u,v)| \le\omega(|t-s|) \Vert u\Vert_{V_\gamma} \Vert v\Vert_{V_\gamma}\quad \ (t,s\in[0,T], u,v\in V),$$ with $$\label{eq 2:Dini-condition}\sup_{t\in[0,T]} \frac{\omega(t)}{t^{\gamma/2}}<\infty \quad \text{ and }
 \int_0^T\frac{\omega(t)}{t^{1+\gamma/2}}<\infty$$ where $V_\gamma:=[H,V]$ is the complex interpolation space. In addition we assume that ${\mathfrak{a}}(t_0; \cdot,\cdot)$ has the square root property for some $t_0\in[0,T]$ (and then for all $t\in[0,T]$ by [@Ar-Mo15 Proposition 2.5] ), i.e.,

$$\label{square property}
D(A^{\frac{1}{2}}(t_0))=V.$$

Recall that for symmetric forms, i.e., if ${\mathfrak{a}}(t;u,v)=\overline{{\mathfrak{a}}(t;v,u)}$ for all $t,u,v,$ then the square root property is satisfied.

Under the assumptions - it is known that for each $x_0\in V$ the non-autonomous homogeneous Cauchy problem $$\label{Abstract Cauchy problem 0}
  \left\{
\begin{aligned}
\dot u(t)&+{\mathcal{A}}(t)u(t)= 0\quad \hbox{a.e. on}\ [0,T],\\
u(0)&=x_0, \,\\
\end{aligned}
\right.$$ has a unique solution $u \in {\textit{MR}\,}(V,H):=L^2(0,T;V)\cap H^1(0,T;H)$ such that $u\in C([0,T];V).$ This result has been proved by Arendt and Monniaux [@Ar-Mo15] (see also [@ELLA15]) when the form ${\mathfrak{a}}$ satisfies the weaker condition $$\label{eq 1:Dini-conditionWeaker} 
  |{\mathfrak{a}}(t,u,v)-{\mathfrak{a}}(s,u,v)| \le\omega(|t-s|) \Vert u\Vert_V \Vert v\Vert_{V_\gamma}\quad \ (t,s\in[0,T], u,v\in V).$$

In this paper we continue to investigate further regularity of the solution of (\[Abstract Cauchy problem 0\]). For this it is necessary to associate to the Cauchy problem an *evolution family* $$\mathcal U:=\Big\{U(t,s):\  0\leq s\leq t\leq T\Big\}\subset {\mathcal{L}}(H)$$ which means that:

-   $U(t,t)=I$ and $U(t,s)= U(t,r)U(r,s)$ for every $0\leq r\leq s\leq t\leq T,$

-   for every $x\in X$ the function $(t,s)\mapsto U(t,s)x$ is continuous into $H$ for $0\leq s\leq t\leq T$

-   for each $x_0\in H, U(\cdot,s)x_0$ is the unique solution of (\[Abstract Cauchy problem 0\]).

Let $Y\subseteq H$ be a subspace. An evolution family $\mathcal U\subset {\mathcal{L}}(H)$ is said to be *norm continuous in $Y$* if $\mathcal U\subset{\mathcal{L}}(Y)$ and the map $(t,s)\mapsto U(t,s)$ is norm continuous with value in ${\mathcal{L}}(Y)$ for $0\leq s<t\leq T.$

If the non-autonomous form ${\mathfrak{a}}$ satisfies the weaker condition (\[eq 1:Dini-conditionWeaker\]) then it is known that (\[Abstract Cauchy problem 0\]) is governed by an evolution family which is norm continuous in $V$ [@LH17 Theorem 2.6], and norm continuous in $H$ if in addition $V\hookrightarrow H$ is compact [@LH17 Theorem 3.4]. However, for many boundary value problem the compactness embedding fails.

In this paper we prove that the compactness assumption can be omitted provided ${\mathfrak{a}}$ satisfies instead of (\[eq 1:Dini-conditionWeaker\]). This will allow us to consider a large class of examples of applications. One of the main ingredient used here is the non-autonomous *returned adjoint form* ${\mathfrak{a}}^*_r : [0,T]\times V\times V{\longrightarrow}{\mathbb{C}}$ defined by $$\label{returnd adjoint form} 
{\mathfrak{a}}^*_r(t,u,v):=\overline{{\mathfrak{a}}(T-t,v,u)} \quad \ (t,s\in[0,T], u,v\in V).$$ The concept of returned adjoint forms appeared in the work of D. Daners [@Daners01] but for different interest. Furthermore, [@LH17 Theorem 2.6] cited above will be also needed to prove our main result.

We note that the study of regularity properties of the evolution family with respect to $(t,s)$ in general Banach spaces has been investigated in the case of constant domains by Komatsu [@H.Ko61] and Lunardi [@Lu89], and by Acquistapace [@Ac88] for time-dependent domains.

We illustrate our abstract results by two relevant examples. The first one concerns the Laplacian with non-autonomous Robin boundary conditions on a unbounded Lipschitz domain. The second one traits a class of Schrödinger operators with time dependent potential.

Preliminary results \[Approximation\]
=====================================

Let $
{\mathfrak{a}}: [0,T]\times V\times V \to {\mathbb{C}}$ a non-autonomous sesquilinear form satisfying and (\[eq:continuity-nonaut\]). Then the following well known result regarding *$L^2$-maximal regularity in $V'$* is due to J. L. Lions

\[wellposedness in V’2\] For each given $s\in[0,T)$ and $x_0\in H$ the homogenuous Cauchy problems $$\label{evolution equation u(s)=x}
\left\{
\begin{aligned}
\dot u(t)&+{\mathcal{A}}(t)u(t)= 0\quad \hbox{a.e. on}\ [s,T],\\
u(s)&=x, \,\\
\end{aligned}
\right.$$ has a unique solution $u \in {\textit{MR}\,}(V,V'):={\textit{MR}\,}(s,T;V,V'):=L^2(s,T;V)\cap H^1(s,T;V').$

Recall that the maximal regularity space ${\textit{MR}\,}(V,V')$ is continuously embedded into $C([s,T],H)$ [@Sho97 page 106]. A proof of Theorem \[wellposedness in V’2\] using a representation theorem of linear functionals, known in the literature as *Lions’s representation Theorem* can be found in [@Sho97 page 112] or [@DL88 XVIII, Chpater 3, page 513].

Furthermore, we consider the non-autonomous adjoint form ${\mathfrak{a}}^*:[0,T]\times V\times V{\longrightarrow}{\mathbb{C}}$ of ${\mathfrak{a}}$ defined by $${\mathfrak{a}}^*(t;u,v):=\overline{{\mathfrak{a}}(t;v,u)}$$ for all $t\in[0,t]$ and $u,v\in V.$ Finally, we will need to consider the returned adjoin form ${\mathfrak{a}}^*_r:[0,T]\times V\times V{\longrightarrow}{\mathbb{C}}$ given by $${\mathfrak{a}}^*_r(t,u,v):={\mathfrak{a}}^*(T-t,u,v).$$ Clearly, the adjoint form is a non-autonomous sesquilinear form and satisfies and (\[eq:continuity-nonaut\]) with the same constant $M, \alpha.$ Moreover, the adjoint operators $A^*(t), t\in[0,T]$ of $A(t), t\in[0,T]$ coincide with the operators associated with ${\mathfrak{a}}^*$ on $H.$ Thus applying Theorem \[wellposedness in V’2\] to the returned adjoint form we obtain that the Cauchy problem associated with ${\mathcal{A}}^*_r(t):={\mathcal{A}}^*(T-t)$ $$\label{evolution equation u(s)=x returned}
\left\{
\begin{aligned}
\dot v(t)&+{\mathcal{A}}^*_r(t)v(t)= 0\quad \hbox{a.e. on}\ [s,T],\\
v(s)&=x, \,\\
\end{aligned}
\right.$$ has for each $x\in H$ a unique solution $v\in MR(V,V').$ Accordingly, for every $(t,s)\in\overline{\Delta}:=\{ (t,s)\in[0,T]^2:\  t\leq s\}$ and every $x\in H$ we can define the following family of linear operators $$\label{evolution family}
 U(t,s)x:=u(t) \quad \hbox{ and }\quad 
U^*_r(t,s)x:=v(t),$$ where $u$ and $v$ are the unique solutions in $MR(V,V')$ respectively of (\[evolution equation u(s)=x\]) and (\[evolution equation u(s)=x returned\]). Thus each family $\{\mathcal U(t,s):\ (t,s)\in\overline{\Delta}\}$ and $\{\mathcal U^*_r(t,s):\ (t,s)\in\overline{\Delta}\}$ yields a contractive, strongly continuous evolution family on $H$ [@LH17 Proposition].

In the autonomous case, i.e., if ${\mathfrak{a}}(t,\cdot,\cdot)={\mathfrak{a}}(\cdot,\cdot)$ for all $t\in[0,T],$ then one knows that $-A_0,$ the operator associated with ${\mathfrak{a}}_0$ in $H,$ generates a $C_0$-semigroup $(T(t))_{t\geq 0}$ in $H.$ In this case ${{U}}(t,s):=T(t-s)$ yields a strongly continuous evolution family on $H.$ Moreover, we have $$\label{equalities: adjoint EVF and EVF autonomous case}
{{U}}(t,s)^\prime=T(t-s)^{\prime}=T^*(t-s)={{U}}^*(t,s)={{U}}^*_r(t,s).$$ Here, $T(\cdot)^\prime$ denote the adjoint of $T(\cdot)$ which coincides with the $C_0$-semigroup $(T^*(t))_{t\geq 0}$ associated with the adjoint form ${\mathfrak{a}}^*.$ In the non-autonomous setting however, (\[equalities: adjoint EVF and EVF autonomous case\]) fails in general even in the finite dimensional case, see [@Daners01 Remark 2.7]. Nevertheless, Proposition \[equalities: adjoint EVF and EVF\] below shows that the evolution families ${{U}}$ and ${{U}}_r^*$ can be related in a similar way. This formula appeared in [@Daners01 Theorem 2.6].

\[equalities: adjoint EVF and EVF\] Let ${{U}}$ and ${{U}}^*_r$ be given by (\[evolution family\]). Then we have $$\label{Key equalities: adjoint EVF and EVF}
    \big [ {{U}}^*_r(t,s)\big ]^\prime x={{U}}(T-s,T-t)x$$ for all $x\in H$ and $(t,s)\in\overline{\Delta}.$

The equality will play a crucial role in the proof of our main result. We include here a new proof for the sake of completeness.

(of Proposition \[equalities: adjoint EVF and EVF\]) Let $\Lambda=(0=\lambda_0<\lambda_1<...<\lambda_{n+1}=T)$ be a subdivision of $[0,T].$ Let ${\mathfrak{a}}_k:V \times V \to \mathbb C\ \ \hbox{ for } k=0,1,...,n$ be given by $$\begin{aligned}
        \ {\mathfrak{a}}_k(u,v):={\mathfrak{a}}_{k,\Lambda}(u,v):=\frac{1}{\lambda_{k+1}-\lambda_k}
        \int_{\lambda_k}^{\lambda_{k+1}}&{\mathfrak{a}}(r;u,v){\rm  d}r\ \hbox{ for } u,v\in V. \
    \end{aligned}$$ All these forms satisfy (\[eq:continuity-nonaut\]) with the same constants $\alpha, M.$ The associated operators in $V'$ are denoted by ${\mathcal{A}}_k\in {\mathcal{L}}(V,V')$ and are given for all $u\in V$ and $k=0,1,...,n$ by $$\label{eq:op-moyen integrale}
{\mathcal{A}}_ku :={\mathcal{A}}_{k,\Lambda}:=\frac{1}{\lambda_{k+1}-\lambda_k}
\int_{\lambda_k}^{\lambda_{k+1}}{\mathcal{A}}(r)u{\rm  d}r.\ \ $$ Consider the non-autonomous form ${\mathfrak{a}}_\Lambda:[0,T]\times V \times V \to {\mathbb{C}}$ defined by $$\label{form: approximation formula1}
    {\mathfrak{a}}_{\Lambda}(t;\cdot,\cdot):=\begin{cases}
    {\mathfrak{a}}_k(\cdot,\cdot)&\hbox{if }t\in [\lambda_k,\lambda_{k+1})\\
    {\mathfrak{a}}_n(\cdot,\cdot)&\hbox{if }t=T\ .
    \end{cases}$$ Its associated time dependent operator ${\mathcal{A}}_\Lambda(\cdot):  [0,T]\subset {\mathcal{L}}(V,V')$ is given by $$\label{form: approximation formula1}
    {\mathcal{A}}_{\Lambda}(t):=\begin{cases}
    {\mathcal{A}}_k&\hbox{if }t\in [\lambda_k,\lambda_{k+1})\\
    {\mathcal{A}}_n &\hbox{if }t=T\ .
    \end{cases}$$ Next denote by $T_k$ the $C_0-$semigroup associated with ${\mathfrak{a}}_k$ in $H$ for all $k=0,1...n.$ Then applying Theorem \[wellposedness in V’2\]) to the form ${\mathfrak{a}}_\Lambda$ we obtain that in this case the associated evolution family ${{U}}_\Lambda(t,s)$ is given explicitly for $\lambda_{m-1}\leq s<\lambda_m<...<\lambda_{l-1}\leq t<\lambda_{l}$ by $$\label{promenade1}{{U}}_\Lambda (t,s):= T_{l-1}(t-\lambda_{l-1})
    T_{l-2}(\lambda_{l-1}-\lambda_{l-2})...T_{m}(\lambda_{m+1}-\lambda_{m})T_{m-1}(\lambda_{m}-s),$$ and for $\lambda_{l-1}\leq a\leq b<\lambda_{l}$ by $$\label{promenade2}U_\Lambda (t,s):= T_{l-1}(t-s).$$ By [@LASA14 Theorem 3.2] we know that $({{U}}_\Lambda)_{\Lambda}$ converges weakly in $MR(V,V')$ as $|\Lambda|\to 0$ and $$\lim\limits_{|\Lambda|\to 0}\|{{U}}_\Lambda-{{U}}\|_{MR(V,V')}=0$$ The continuous embedding of $MR(V,V')$ into $C([0,T];H)$ implies that $\lim\limits_{|\Lambda|\to 0}{{U}}_\Lambda={{U}}$ in the weak operator topology of ${\mathcal{L}}(H).$

Now, let $(t,s)\in \overline{\Delta}$ with $\lambda_{m-1}\leq s<\lambda_m<...<\lambda_{l-1}\leq t<\lambda_{l}$ be fixed. Applying the above approximation argument to ${\mathfrak{a}}_r^*$ one obtains that $$\begin{aligned}
\label{eq1 proof returned adjoint}
{{U}}^*_{\Lambda,r}(t,s)&=T_{l-1,r}^*(t-\lambda_{l-1})
T_{l-2,r}^*(\lambda_{l-1}-\lambda_{l-2})...T_{m,r}^*(\lambda_{m+1}-\lambda_{m})T_{m-1,r}^*(\lambda_{m}-s)
\\\label{eq2 proof returned adjoint}&=T_{l-1,r}^\prime(t-\lambda_{l-1})
T_{l-2,r}^\prime(\lambda_{l-1}-\lambda_{l-2})...T_{m,r}^\prime(\lambda_{m+1}-\lambda_{m})T_{m-1,r}^\prime(\lambda_{m}-s)
$$ where $T_{k,r}$ and $T_{k,r}^*$ are the $C_0$-semigroups associated with $$\label{eq1 proof Thm equalities: adjoint EVF and EVF} {\mathfrak{a}}_{k,r}(u,v):=\frac{1}{\lambda_{k+1}-\lambda_k}\int_{\lambda_k}^{\lambda_{k+1}}{\mathfrak{a}}(T-r;u,v){\rm  d}r=\frac{1}{\lambda_{k+1}-\lambda_k}\int_{T-\lambda_{k+1}}^{T-\lambda_k}{\mathfrak{a}}(r;u,v){\rm  d}r$$ and its adjoint ${\mathfrak{a}}_{k,r}^*$, respectively. Recall that $T_{k,r}^*=T_{k,r}^\prime.$

On the other hand, the last equality in (\[eq1 proof Thm equalities: adjoint EVF and EVF\]) implies that $T_{k,r}$ coincides with the semigroup associated with ${\mathfrak{a}}_{k,\Lambda_T}$ where $\Lambda_T$ is the subdivision $\Lambda_T:=(0=T-\lambda_{n+1}<T-\lambda_n<...<T-\lambda_1<T-\lambda_0=T).$ It follows from - and - that $$\begin{aligned}
\Big[{{U}}^*_{\Lambda,r}(t,s)\Big]^\prime&={\mathcal{T}}_{m-1,r}(\lambda_{m}-s){\mathcal{T}}_{m,r}(\lambda_{m+1}-\lambda_{m})...{\mathcal{T}}_{l-2,r}(\lambda_{l-1}-\lambda_{l-2}){\mathcal{T}}_{l-1,r}(t-\lambda_{l-1})
\\&={\mathcal{T}}_{m-1}\Big((T-s)-(T-\lambda_{m})\Big){\mathcal{T}}_{m}\Big((T-\lambda_m)-(T-\lambda_{m+1})\Big)...{\mathcal{T}}_{l-1}\Big((T-\lambda_{l-1})-(T-t)\Big)\\&={{U}}_{\Lambda_T}(T-s,T-t)\end{aligned}$$ Finally, the desired equality follows by passing to the limit as $|\Lambda|=|\Lambda_T|\to 0.$

\[remark-rescaling\] The coerciveness assumption in may be replaced with $$\label{eq:Ellipticity-nonaut2}
    {\operatorname{Re}}{\mathfrak{a}}(t,u,u) +\omega\Vert u\Vert_H^2\ge \alpha \|u\|^2_V \quad ( t\in [0,T], u\in V)$$ for some $\omega\in {\mathbb{R}}.$ In fact, ${\mathfrak{a}}$ satisfies (\[eq:Ellipticity-nonaut2\]) if and only if the form $a_\omega$ given by ${\mathfrak{a}}_\omega(t;\cdot,\cdot):={\mathfrak{a}}(t;\cdot,\cdot)+\omega (\cdot{\, \vert \,}\cdot)$ satisfies the second inequality in . Moreover, if $u\in MR(V,V')$ and $v:=e^{-w.}u,$ then $v\in MR(V,V')$ and $u$ satisfies (\[evolution equation u(s)=x\]) if and only if $v$ satisfies $$\dot{v}(t)+(\omega+\mathcal A(t))v(t)=0 \ \ \  t{\rm
        -a.e.}  \hbox{ on} \ [s,T],\
    \ \ \ \ v(s)=x. $$

Norm continuous evolution family {#Sec2 Norm continuity}
================================

In this section we assume that the non-autonomous form ${\mathfrak{a}}$ satisfies (\[eq:continuity-nonaut\])-. Thus as mentioned in the introduction, under theses assumptions the Cauchy problem (\[evolution equation u(s)=x\]) has $L^2$-maximal regularity in $H$. Thus for each $x\in V,$ $$U(\cdot,s)x\in {\textit{MR}\,}(V,H):={\textit{MR}\,}(s,T;V,H):=L^2(s,T;V)\cap H^1(s,T;H).$$ Moreover, $ U(\cdot,s)x\in C[s,T];V)$ by [@Ar-Mo15 Theorem 4.2]. From [@LH17 Theorem 2.7] we known that the restriction of ${{U}}$ to $V$ defines an evolution family which norm continuous. The same is also true for the Cauchy problem (\[evolution equation u(s)=x returned\]) and the assocaited evolution family $U^*_r$ since the returned adjoint form ${\mathfrak{a}}_r^*$ inherits all properties of ${\mathfrak{a}}.$

In the following we establish that ${{U}}$ can be extended to a strongly continuous evolution family on $V'.$

\[Lemma EVF on V’\] Let ${\mathfrak{a}}$ be a non-autonomous sesquilinear form satisfying (\[eq:continuity-nonaut\])-(\[square property\]). Then ${{U}}$ can be extended to a strongly continuous evolution family on $V^\prime,$ which we still denote ${{U}}.$

Let $x\in H$ and $(t,s)\in\overline {\Delta}.$ Then using Proposition \[equalities: adjoint EVF and EVF\] and the fact that ${{U}}$ and ${{U}}_r^*$ define both strongly continuous evolution families on $V$ and $H$ we obtain that $$\begin{aligned}
\|{{U}}(t,s)x\|_{V'}&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}<{{U}}(t,s)x, v>{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}({{U}}(t,s)x|v)_H{\, \vert \,}=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}(x| {{U}}(t,s)^\prime v)_H{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}(x|{{U}}_r^*(T-s,T-t)v)_H{\, \vert \,}\\&=\sup_{\underset{v\in V}{\|v\|_V=1}}{\, \vert \,}<x,{{U}}_r^*(T-s,T-t)v>{\, \vert \,}\\&\leq \|x\|_{V^\prime}\|{{U}}_r^*(T-s,T-t)\|_{{\mathcal{L}}(V)}
\\&\leq c\|x\|_{V^\prime}\end{aligned}$$ where $c>0$ is such that $\underset{t,s\in\Delta}{\sup}\|{{U}}_r^*(t,s)\|_{{\mathcal{L}}(V)}\leq c.$ Thus, the claim follows since $H$ is dense in $V'.$

Let $\Delta:=\{(t,s)\in\Delta{\, \vert \,}t\ge s\}.$ The following theorem is the main result of this paper

\[main result\] Let ${\mathfrak{a}}$ be a non-autonomous sesquilinear form satisfying (\[eq:continuity-nonaut\])-(\[square property\]). Let $\{U(t,s):\ (t,s)\in\Delta\}$ given by (\[evolution family\]). Then the function $(t,s)\mapsto {{U}}(t,s)$ is norm continuous on $\Delta$ into ${\mathcal{L}}(X)$ for $X=V, H$ and $V'.$

The norm continuity for ${{U}}$ in the case where $X=V$ follows from [@LH17 Theorem 2.7]. On the other hand, applying [@LH17 Theorem 2.7] to ${\mathfrak{a}}_r^*$ we obtain that ${{U}}_r^*$ is also norm continuous on $\Delta$ with values in ${\mathcal{L}}(V).$ Using Proposition \[equalities: adjoint EVF and EVF\], we obtain by similar arguments as in the proof of Lemma \[Lemma EVF on V’\] $$\|{{U}}(t,s)-{{U}}(t',s')\|_{{\mathcal{L}}(V')}\leq \|{{U}}_r^*(T-s,T-t)x-{{U}}_r^*(T-s',T-t')x\|_{{\mathcal{L}}(V)}$$ for all $x\in V'$ and $(t,s), (t',s')\in \Delta.$ This implies that ${{U}}$ is norm continuous on $\Delta$ with values in ${\mathcal{L}}(V').$ Finally, the norm continuity in $H$ follows then by interpolation.

Examples {#S application}
========

This section is devoted to some relevant examples illustrating the theory developed in the previous sections. We refer to [@Ar-Mo15] and [@Ou15] and the references therein for further examples.

$(i)$ **Laplacian with time dependent Robin boundary conditions on exterior domain** Let $\Omega$ be a bounded domain of ${\mathbb{R}}^d$ with Lipschitz boundary $\Gamma.$ Denote by $\sigma$ the $(d-1)$-dimensional Hausdorff measure on $\Gamma.$ Let $\Omega_{ext}$ denote the exterior domain of $\Omega,$ i.e., $\Omega_{ext}:={\mathbb{R}}^d\setminus\overline{\Omega}.$ Let $T>0$ and $\alpha>1/4.$ Let $\beta:[0,T]\times \Gamma {\longrightarrow}{\mathbb{R}}$ be a bounded measurable function such that $$|\beta(t,\xi)-\beta(t,\xi)|\leq c|t-s|^\alpha$$ for some constant $c>0$ and every $t,s\in [0,T], \xi\in \Gamma.$ We consider the from ${\mathfrak{a}}:[0,T]\times H^1(\Omega_{ext})\times H^1(\Omega_{ext}){\longrightarrow}{\mathbb{C}}$ defined by $${\mathfrak{a}}(t;u,v):=\int_{\Omega_{ext}}\nabla u\cdot\nabla v {\rm d}\xi+\int_{\Omega_{ext}}\beta(t,\cdot){{u}{_{|\Gamma}}} {{\bar v}{_{|\Gamma}}} {\rm d}\sigma$$ where $u\to {{u}{_{|\Gamma}}}: H^1(\Omega_{ext}) {\longrightarrow}L^2(\Gamma,\sigma)$ is the trace operator which is bounded [@AdFou Theorem 5.36]. The operator $A(t)$ associated with ${\mathfrak{a}}(t;\cdot,\cdot)$ on $H:=L^2(\Omega_{ext})$ is minus the Laplacian with time dependent Robin boundary conditions $$\partial_\nu u(t)+\beta(t,\cdot)u=0\ \text{ on } \Gamma.$$ Here $\partial_\nu$ is the weak normal derivative. Thus the domain of $A(t)$ is the set $$D(A(t))=\Big\{ u\in H^1(\Omega_{ext}) {\, \vert \,}{\mathop{}\!\mathbin\bigtriangleup}u\in L^2(\Omega_{ext}), \partial_\nu u(t)+\beta(t,\cdot){{u}{_{|\Gamma}}}=0 \Big\}$$ and for $u\in D(A(t)), A(t)u:=-{\mathop{}\!\mathbin\bigtriangleup}u.$ Thus similarly as in [@Ar-Mo15 Section 5] one obtains that ${\mathfrak{a}}$ satisfies (\[eq:continuity-nonaut\])-(\[square property\]) with $\gamma:=r_0+1/2$ and $\omega(t)=t^\alpha$ where $r_0\in(0,1/2)$ such that $r_0+1/2<2\alpha.$ We note that [@Ar-Mo15 Section 5] the authors considered the Robin Laplacian on the bounded Lipschitz domain $\Omega.$ The main ingredient used there is that the trace operators are bounded from $H^{s}(\Omega)$ with value in $H^{s-1/2}(\Gamma,\sigma)$ for all $1/2<s<3/4.$ This boundary trace embedding theorem holds also for unbounded Lipschitz domain, and thus for $\Omega_{ext}$, see [@Mclean Theorem 3.38] or [@Cos Lemma 3.6].

Thus applying [@Ar-Mo15 Theorem 4.1] and Theorem \[main result\] we obtain that the non-autonomous Cauchy problem

$$\label{RobinLpalacian}
 \left\{
 \begin{aligned}
 \dot {u}(t) - {\mathop{}\!\mathbin\bigtriangleup}u(t)& = 0, \ u(0)=x\in H^1(\Omega_{ext })
 \\  \partial_\nu u(t)+\beta(t,\cdot){u}&=0 \ \text{ on } \Gamma
 \end{aligned} \right.$$

has $L^2$-maximal regularity in $L^2(\Omega_{ext})$ and its solution is governed by an evolution family ${{U}}(\cdot,\cdot)$ that is norm continuous on each space $V, L^2(\Omega_{ext})$ and $V'.$

Non-autonomous Schrödinger operators
------------------------------------

Let $m_0, m_1\in L_{Loc}^1({\mathbb{R}}^d)$ and $m:[0,T]\times{\mathbb{R}}^d{\longrightarrow}{\mathbb{R}}$ be a measurable function such that there exist positive constants $\alpha_1,\alpha_2$ and $\kappa$ such that $$\alpha_1 m_0(\xi)\leq m(t,\xi)\leq \alpha_2 m_0(\xi),
 \quad \text{ }\
 \quad {\, \vert \,}m(t,\xi)-m(s,\xi){\, \vert \,}\leq \kappa|t-s|m_1(\xi)$$ for almost every $\xi\in {\mathbb{R}}^d$ and every $t,s\in [0,T].$ Assume moreover that there exist a constant $c>0$ and $s\in[0,1]$ such that for $u\in C_c^\infty({\mathbb{R}}^d)$ $$\label{additional AS Svhrödinger example}\int_{{\mathbb{R}}^d}m_1(\xi)|u(\xi)|^2 {\mathrm{d}}\xi\leq c\|u\|_{H^s({\mathbb{R}}^d)}.$$Consider the non-autonomous Cauchy problem $$\label{Schroedinger operator}
\left\{
\begin{aligned}
\dot {u}(t) - &{\mathop{}\!\mathbin\bigtriangleup}u(t)+m(t,\cdot)u(t) = 0,
\\  u(0)&=x\in V. 
\end{aligned} \right.$$ Here $A(t)=-{\mathop{}\!\mathbin\bigtriangleup}+m(t,\cdot)$ is associated with the non-autonomous form ${\mathfrak{a}}:[0,T]\times V\times V{\longrightarrow}{\mathbb{C}}$ given by $$V:=\left\{u\in H^1({\mathbb{R}}^d):  \int_{{\mathbb{R}}^d}m_0(\xi)|u(\xi)|^2  {\mathrm{d}}\xi <\infty \right\}$$and $${\mathfrak{a}}(t;u,v)=\int_{{\mathbb{R}}^d}\nabla u\cdot\nabla v  {\rm d} \xi+\int_{{\mathbb{R}}^d}m(t,\xi)|u(\xi)|^2 {\mathrm{d}}\xi.$$ The form ${\mathfrak{a}}$ satisfies also (\[eq:continuity-nonaut\])-(\[square property\]) with $\gamma:=s$ and $\omega(t)=t^\alpha$ for $\alpha>\frac{s}{2}$ and $s\in[0,1].$

This example is taken from [@Ou15 Example 3.1]. Using our Theorem \[main result\] we have that the solution of Cauchy problem (\[Schroedinger operator\]) is governed by a norm continuous evolution family on $L^2({\mathbb{R}}^d), V$ and $V'.$

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[^1]: $^*$This work is supported by Deutsche Forschungsgemeinschaft DFG (Grant LA 4197/8-1)
