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 , and the classical bounded slope condition on the initial and the lateral boundary datum . We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – 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 .
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. 355–379DOI 10.1016/J.ANIHPC.2015.12.005