Parabolic equations and the bounded slope condition
Frank Duzaar, Paolo Marcellini, Stefano Signoriello, Verena Bögelein · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2015
In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations \begin{cases} \partial _{t}u−\mathrm{div}\:Df(Du) = 0 &\text{in }\mathrm{\Omega }_{T}\text{,} \\ u = u_{o} &\text{on }\partial _{\mathcal{P}}\mathrm{\Omega }_{T}\text{.} \end{cases} 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} .