Non local diffusion equations - Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo; Signoriello, Stefano (Journal of Mathematical Analysis and Applications, 2015)
Keywords: Existence of solutions; Parabolic equations; Variational minimizer; Parabolic minimizer; Non local partial differential equations

Parabolic equations and the bounded slope condition - Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo; Signoriello, Stefano (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2017)
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.<br