Parabolic PDEs with Dynamic Data under a Bounded Slope Condition
Verena Bögelein, Frank Duzaar, Giulia Treu
TL;DR
The paper develops a Haar-type theory for parabolic equations with time-dependent boundary data under a time-dependent bounded slope condition. It introduces a time-adaptive barrier method based on the convex conjugate $f^*$ and a De Giorgi–type minimizing movement scheme to construct Lipschitz spatial solutions, even when lower-order terms are present. Existence and uniqueness of variational solutions with a uniform spatial Lipschitz bound are established, and, for $f∈C^1$, the solutions are weak solutions with strong regularity like $u∈C^{0;1,1/2}(Ω_T)$. The results extend classical stationary Lipschitz minimizers to non-autonomous boundary data, providing explicit barrier constructions and robust gradient control via a parabolic comparison framework.
Abstract
We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_ξf(\nabla u)=0 $$ in a space-time cylinder $Ω_T=Ω\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}Ω_T\to \mathbf{R}$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data $g$ along the lateral boundary $\partialΩ\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partialΩ$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.