Gradient bounds for a widely degenerate orthotropic parabolic equation

TL;DR

This paper proves local Lipschitz continuity of solutions to a degenerate orthotropic parabolic equation, extending previous elliptic and less degenerate results to a more complex, highly degenerate setting.

Contribution

It establishes the first Lipschitz regularity results for solutions of a strongly degenerate, orthotropic parabolic PDE, bridging elliptic theory and advanced degeneracy.

Findings

Solutions are locally Lipschitz continuous in space

The result extends previous elliptic regularity to a parabolic setting

Addresses a highly degenerate, anisotropic PDE framework

Abstract

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega\times I, \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interval, $p\geq2$, $\delta_{1},\ldots,\delta_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. We prove that the local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. We emphasize that our result can be considered, on the one hand, as the parabolic counterpart of the elliptic result established in [12], and on the other hand as an extension to a significantly more degenerate framework of the findings contained in [13].