A proof of the Cp′-regularity conjecture in the plane

Abstract

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C p ′ = C 1, 1 p−1 ; this regularity is optimal.

An asymptotic sharp Sobolev regularity for planar infinity harmonic functions

“…Remark 1.8. It would be interesting to show that Theorem 1.6, and hence Corollary 1.7, holds for p ∈ [1,2].…”

“…Also it is related to the hodograph transformation, which, for example, was applied to show the sharp Hölder regularity of solutions to certain equations involving the p-Laplacian; see e.g. [19] and [1]. We also refer to [8,Chapter 16] for more applications of hodograph transformation.…”

Sharp regularity estimates for quasilinear evolution equations

“…We then iterate this estimate in a systematic manner, properly adjusted to the intrinsic scaling of equation. Inspired by the recent results from [5,6,8] and [39], we obtain an estimate (Theorem 3.2), which provides a precise control of the oscillation of weak solution of (1.1) in terms of the magnitude of its gradient.…”

“…[10] and [25] for complete essays on regularity of evolution equations with degenerate diffusion). Although weak solutions of (1.1) under the compatibility assumptions (C) are known to be locally of the class C 1+α (in the parabolic sense) for some α ∈ (0, 1), the sharp exponent is known only for some specific cases (see [5,6,26,28,33]). This type of quantitative information plays an essential role in the study of blow-up analysis, related geometric and free boundary problems and for proving Liouville type results (see [3,4,12,16,18,40] for some enlightening examples).…”

A note on the optimal boundary regularity for the planar generalized p-Poisson equation

“…In [15], (1) was studied with f ∈ L q , 2 < q < ∞ and optimal interior regularity was achieved in plane by Lindgren and Lindqvist. Recently in an interesting work of Araujo and Zhang, [2] more general p-Poisson equation (but h = 0) is studied and some interior regularity is achieved. We assume A ∈ C (1−2/q)/(p−1) to achieve the same regularity as in the case of the p-Poisson equation.…”