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

TL;DR

This paper proves the existence and uniqueness of Lipschitz continuous solutions for a class of parabolic equations with time-dependent integrands, under bounded slope conditions on initial and boundary data.

Contribution

It establishes the bounded slope condition as sufficient for well-posedness and regularity of solutions to parabolic equations with time-dependent convex integrands.

Findings

Existence of a unique variational solution.

Solution is Lipschitz continuous in space.

Bounded slope condition ensures regularity.

Abstract

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