Parabolic equations and the bounded slope condition

In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations{∂tu−divDf(Du)=0in ΩT,u=uoon ∂PΩT. The only assumptions needed are the convexity of the generating function f:Rn→R, and the classical bounded slope condition on the initial and the lateral boundary datum uo∈W1,∞(Ω). We emphasize that no growth conditions are assumed on f and that - an example which does not enter in the elliptic case - uo could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in Rn.