Parabolic equations and the bounded slope condition

Abstract

In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations

The only assumptions needed are the convexity of the generating function $f:{\mathbb{R}}^n\to \mathbb{R}$, and the classical bounded slope condition on the initial and the lateral boundary datum $u_o\in W^{1,\infty }(\mathrm{\Omega })$. We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – $u_o$ 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 ${\mathbb{R}}^n$.