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
Parabolic equations and the bounded slope condition cover
Download PDF

Abstract

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

{tudivDf(Du)=0in ΩT,u=uoon PΩT.\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:RnRf:\mathbb{R}^{n}\rightarrow \mathbb{R}, and the classical bounded slope condition on the initial and the lateral boundary datum uoW1,(Ω)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 – uou_{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 Rn\mathbb{R}^{n}.

Cite this article

Frank Duzaar, Paolo Marcellini, Stefano Signoriello, Verena Bögelein, Parabolic equations and the bounded slope condition. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 355–379

DOI 10.1016/J.ANIHPC.2015.12.005