Parabolic equations and the bounded slope condition

In this talk we discuss the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations \begin{equation*} \left\{ \begin{array}{c} \partial_tu- div Df(Du)=0\quad\mbox{in $\Omega\times(0,T)$,}\\[3pt] u=u_o\quad\mbox{on $(\partial\Omega\times(0,T))\cup(\overline\Omega\times\{0\})$.} \end{array} \right. \end{equation*} One prominent example which is included in this framework is the time dependent minimal surface equation. The only assumptions needed are the convexity of the generating function $f\colon\ \mathbb R^n \to \mathbb R$, and the classical bounded slope condition on the initial and the lateral boundary datum $u_o\in W^{1,\infty}(\Omega)$. We emphasize that no growth conditions are assumed on $f$.