JournalsaihpcVol. 34, No. 2pp. 355–379

# Parabolic equations and the bounded slope condition

• ### Frank Duzaar

Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
• ### Paolo Marcellini

Dipartimento di Matematica e Informatica “U.Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
• ### Stefano Signoriello

Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
• ### Verena Bögelein

Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria ## Abstract

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

$\left\{\begin{matrix} \partial _{t}u−\mathrm{div}\:Df(Du) = 0\:\text{in }\mathrm{\Omega }_{T}\text{,} \\ u = u_{o}\:\text{on }\partial _{\mathscr{P}}\mathrm{\Omega }_{T}\text{.} \\ \end{matrix}\right.$

The only assumptions needed are the convexity of the generating function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, 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 $\mathbb{R}^{n}$.