Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

Frank Duzaar (Salzburg)

Abstract: We consider the Cauchy–Dirichlet problem for a class of nonlinear parabolic equations of the form
\begin{equation*}
\partial_tu-\mathrm{div}_x \nabla_\xi f(\nabla u)=0
\end{equation*}
in a space-time cylinder $\Omega_T=\Omega\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}\Omega_T\to \mathbb R$ prescribed on the parabolic boundary. We establish the existence of Lipschitz continuous solutions under minimal regularity assumptions on the data.

The principal novelty of the present work is the introduction of a time-dependent variant of the classical bounded slope condition. Specifically, we assume that for each fixed time $t\in [0,T]$, the spatial trace $g(\cdot,t)$ admits supporting hyperplanes along $\partial\Omega$ with slopes that may vary in time but remain uniformly bounded. This geometric condition is sufficiently flexible to accommodate genuinely time-dependent boundary values.
The proof is based on the construction of upper and lower barriers tailored to the parabolic geometry, which constitutes a central innovation in our method.

This work was carried out in collaboration with Giulia Treu (University of Padova) and Verena Bögelein (University of Salzburg).