Gradient bounds for a widely degenerate orthotropic parabolic equation
Pasquale Ambrosio
TL;DR
The paper addresses gradient regularity for the strongly degenerate, anisotropic parabolic equation $\partial_{t}u=\sum_{i=1}^{n}\partial_{x_{i}}\big[(|u_{x_{i}}|-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{|u_{x_{i}}|}\big]$ and proves local Lipschitz continuity of local weak solutions in space, uniformly in time. The authors introduce a regularized problem with a smooth, uniformly convex potential $F_{\varepsilon}$, derive a priori Lipschitz bounds via a double Caccioppoli framework and a Moser iteration performed with a measure $\mathrm{d}\mu$ that localizes away from degeneracy, and then pass to the limit as $\varepsilon\to0$ to obtain the result for the original equation. The analysis extends elliptic orthotropic results to the parabolic setting and handles the significant degeneracy when $\max_i\delta_i>0$, providing quantitative $L^{\infty}$ control of the gradient and robust energy estimates. The work thus furnishes a parabolic counterpart to known elliptic theories, with potential impact on the study of highly anisotropic degenerate diffusion processes.
Abstract
In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω\times I, \] where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interval, $p\geq2$, $δ_{1},\ldots,δ_{n}$ 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].