Evolution equations governed by Lipschitz continuous non-autonomous forms

summary:We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy \rlap {problem} $$ \dot {u} (t)+A(t)u(t)=f(t) \quad \text {for a.e.\ } t\in [0,T],\quad u(0)=u_0, $$ where the operator $A(t)$ arises from a time depending sesquilinear form $\mathfrak {a}(t,\cdot ,\cdot )$ 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 by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of $H.