Sharp Sobolev regularity for widely degenerate parabolic equations

TL;DR

$\partial_t u-\operatorname{div}( (|Du|-\lambda)_+^{p-1}\frac{Du}{|Du|} )=f$ with $p\ge2$ and $\lambda\ge0$ is shown to admit Sobolev spatial regularity for a nonlinear gradient function $\mathcal{V}_{\alpha,\lambda}(Du)$ under data $f$ in Lebesgue-Besov parabolic spaces when $p>2$, and in $L^2_{\text{loc}}$ when $p=2$. The method combines a regularization scheme with a priori estimates and a Besov-space duality framework to handle $f$ with limited spatial regularity, and then transfers the regularity from the regularized problems to the original solution through a comparison argument. The results yield the existence of a weak time derivative for the evolutionary $p$-Poisson equation and extend elliptic higher differentiability phenomena to a strongly degenerate parabolic setting, with optimal Besov-type assumptions on $f$. The work thus links parabolic regularity theory with Besov-space techniques to accommodate degeneracy and provides a parabolic analogue of known elliptic results, offering sharp spatial regularity and time-derivative conclusions under minimal regularity on the data.

Abstract

We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ Ω_{T}=Ω\times(0,T), \] where $p\geq2$, $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a non-negative constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Besov parabolic space when $p>2$ and that $f\in L_{loc}^{2}(Ω_{T})$ if $p=2$, we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary $p$-Poisson equation. The main novelty here is that $f$ only has a Besov or Lebesgue spatial regularity, unlike the previous work [6], where $f$ was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [5], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations.

Theorems & Definitions (46)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5