  \begin{abstract}\label{abstract} We consider a non-autonomous evolutionary
problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator
$\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$
and $u_0\in V$. Our main concern is to prove well-posedness with maximal
regularity which means the following. Given a Hilbert space $H$ such that $V$
is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$ we
are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove
well-posedness in this sense whenever the form is piecewise
Lipschitz-continuous and symmetric. Moreover, we show that each solution is in
$C([0,T];V)$. We apply the results to non-autonomous Robin-boundary conditions
and also use maximal regularity to solve a quasilinear problem.
