Evolution Equations governed by Lipschitz Continuous Non-autonomous Forms

We prove $L^2$-maximal regularity of linear non-autonomous evolutionary Cauchy problem \begin{equation}\label{eq00}\nonumber \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time dependent sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed in \cite{ELKELA11}, \cite{ELLA13} and \cite{LH}. As a consequence, we obtain an invariance criterion for convex and closed sets of $H.$