Parabolic equations and the bounded slope condition

Bögelein V, Duzaar F, Marcellini P, Signoriello S (2017)

Publication Type: Journal article

Publication year: 2017

Journal

Annales de l'Institut Henri Poincaré - Analyse Non Linéaire Elsevier Masson / Institute Henri Poincaré

Book Volume: 34

Pages Range: 355--379

Journal Issue: 2

DOI: 10.1016/j.anihpc.2015.12.005

Abstract

In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations ∂tu − div Df (Du) = 0 in T, u = uo on ∂PT. 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,∞(). 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 ∂, and the boundary may contain flat parts, for instance could be a rectangle in Rn.

Authors with CRIS profile

Frank Duzaar Lehrstuhl für Partial Differential Equations Stefano Signoriello

Involved external institutions

Università degli Studi di Firenze / University of Florence

Italy (IT) Universität Salzburg (Paris Lodron Universität Salzburg)

Austria (AT)

How to cite

APA:

Bögelein, V., Duzaar, F., Marcellini, P., & Signoriello, S. (2017). Parabolic equations and the bounded slope condition. Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 34(2), 355--379. https://doi.org/10.1016/j.anihpc.2015.12.005

MLA:

Bögelein, Verena, et al. "Parabolic equations and the bounded slope condition." Annales de l'Institut Henri Poincaré - Analyse Non Linéaire 34.2 (2017): 355--379.

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