The bounded slope condition for parabolic equations with time-dependent integrands

Abstract

In this paper, we study the Cauchy-Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \mbox{${\partial }_{t}u-\mathrm{div}\left(D_{\xi }f(t,Du)\right)=0$ } & \mbox{in ${\Omega }_T$}, \\[5pt] \mbox{$u=u_o$} & \mbox{on ${\partial }_{\mathcal{P}}{\Omega }_T$},\\[5pt] \end{array} \right. \end{equation*} where $\Omega \subset {\mathbb{R}}^n$ is a convex domain, $f:[0,T]\times {\mathbb{R}}^n\to \mathbb{R}$ is $L^1$-integrable in time and convex in the second variable. Assuming that the initial and boundary datum $u_o:\overline{\Omega }\to \mathbb{R}$ satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.

26 pages