The one-sided bounded slope condition in evolution problems

Abstract

We establish a local Lipschitz regularity result of solutions to the Cauchy–Dirichlet problem associated with evolutionary partial differential equations ∂tu-divDf(∇u)=0,inΩT,u=u0,on∂PΩT.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _t u - {{\\,\\mathrm{div}\\,}}Df(\\nabla u) = 0, &{} \\quad \\text{ in } \\, \\Omega _T,\\\\ u=u_0, &{} \\quad \\text{ on } \\, \\partial _{{\\mathcal {P}}}\\, \\Omega _T. \\end{array} \\right. \\end{aligned}$$\\end{document}We do not impose any growth assumptions from above on the function f :{mathbb {R}}^n rightarrow {mathbb {R}} and only require it to be convex and coercive. The domain Omega is required to be bounded and convex, and the time-independent boundary datum u_0 is supposed to be convex and Lipschitz continuous on {overline{Omega }}. It can be seen as an evolutionary analogue to the one-sided bounded slope condition. Additionally, assuming Omega to be uniformly convex, we establish global continuity on overline{Omega _T} of the solution.