Qualitative-to-quantitative regularity for degenerate equations
Abstract
We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the p-Laplace equation, p>2, we prove that local C1,γ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform C1,β estimates for every 0<β<γ. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the Lp norm of the solution for every exponent below the sharp Iwaniec--Manfredi threshold. Applied to Xu's recent qualitative C1,1/3 regularity for planar infinity-harmonic functions, we obtain uniform interior C1,β estimates for every 0<β<1/3. For the nonlocal superconductivity equation |Du|(−Δ)s u=0, with 1/2<s<1, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior C1,α estimates for bounded solutions, for some α>0. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.
Related
- Global regularity for a class of degenerate/singular Monge-Ampère equations
- Local regularity for solutions to singular and degenerate quasilinear parabolic equations
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- Regularity for solutions to inhomogeneous degenerate parabolic p-Laplace equations
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
BibTeX
@misc{sobral2026qualitativetoquantitative,
title = {{Qualitative-to-quantitative regularity for degenerate equations}},
author = {Aelson Sobral and Eduardo V. Teixeira and José Miguel Urbano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13137},
eprint = {2610.13137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13137},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Qualitative-to-quantitative regularity for degenerate equations
AU - Aelson Sobral
AU - Eduardo V. Teixeira
AU - José Miguel Urbano
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13137
UR - https://arxiv.org/abs/2610.13137
AB - We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the p-Laplace equation, p>2, we prove that local C^1,γ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform C^1,β estimates for every 0<β<γ. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the L^p norm of the solution for every exponent below the sharp Iwaniec--Manfredi threshold. Applied to Xu's recent qualitative C^1,1/3 regularity for planar infinity-harmonic functions, we obtain uniform interior C^1,β estimates for every 0<β<1/3. For the nonlocal superconductivity equation |Du|(−Δ)^s u=0, with 1/2<s<1, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior C^1,α estimates for bounded solutions, for some α>0. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.
ER -