Hölder–Zygmund estimates for degenerate parabolic systems

Abstract

We consider energy solutions of the inhomogeneous parabolic p-Laplacien system ∂ t u − div(|∇u| p−2 ∇u) = −divg). We show in the case p ≥ 2 that if the right hand side g is locally in L ∞ (BMO), then u is locally in L ∞ (C 1 ), where C 1 is the 1-Hölder-Zygmund space. This is the borderline case of the Calderón-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of g are conserved by ∇u, e.g. Hölder continuity. Moreover, we prove a new decay for gradients of p-caloric solutions for all 2n n+2 < p < ∞.

Higher integrability for the singular porous medium system

“…Construction of a non-uniform system of cylinders. The following construction of a non-uniform system of cylinders is similar to the one in [11,19]. Let z o ∈ Q 2R .…”

Self-improving property of degenerate parabolic equations of porous medium-type

“…It is noteworthy that, based on the local higher integrability result of the parabolic p-Laplacian, many applications follow. These include (without any ambition of completion) the full L q theory and beyond [AM07,Sch13], as well as partial regularity results [BZM13], or pointwise estimates via potential theory for equations [KM14/1]. Summarizing, many different ways to show regularity for the gradient of solutions to the p-Laplacian are available which, among other benefits, has a natural impact on the regularity of the time derivative, as it was recently shown in [FSch15].…”

“…, we have to develop something like an intrinsic metric with respect to u, which can then be used for the Calderón-Zygmund analysis, independently of the PDE. In turn, this heavily relies on the possibility of adapting the construction of [Sch13] to the porous medium equation.…”

Regularity for parabolic systems of Uhlenbeck type with Orlicz growth

“…The local L ∞ -bound (of DiBenedetto and Friedman) of the gradient of p-caloric functions was successfully used to derive higher integrability results [1]. The Hölder estimates for the gradients of p-caloric functions has been used to show Hölder continuity for the inhomogeneous system [12,24] and to derive estimates of BMO-Type [27]. Moreover, it is a necessary tool for the proof of pointwise potential estimates [19,20] and for almost everywhere regularity results by p-caloric approximation [7].…”