Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
Abstract
This is a continuation of our recent work~\cite{WZJ26} on asymptotic gradient regularity for nonlinear parabolic equations. In this paper, we consider bounded viscosity solutions of the degenerate or singular fully nonlinear parabolic equation ut−|Du|γF(D2u)=f in Q1, where $ γ>-1 $, $ F $ is uniformly $ (λ, Λ)$-elliptic with 0<λ≤Λ, F(0)=0, and f∈ C0(Q1)∩ L∞(Q1). Given any prescribed α∈(0,1), we prove that there exists $ \varepsilon=\varepsilon(n,α)>0 $ such that |γ|+(Λ/λ−1) ≤ ε Longrightarrow u ∈ Cloc1+α,(1+α)/2(Q1). In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $1$ as $ (γ, \fracΛλ) \to (0,1) $. This provides a fully nonlinear counterpart to the asymptotic gradient regularity result established in~\cite{WZJ26}.
Related
- Local regularity for solutions to singular and degenerate quasilinear 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
- Homogenization of degenerate parabolic equations with nonlinear Robin reactions in perforated domains
- Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
BibTeX
@misc{wang2026asymptotic,
title = {{Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations}},
author = {Jiangwen Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23683},
eprint = {2610.23683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23683},
}
RIS
TY - UNPB
TI - Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
AU - Jiangwen Wang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23683
UR - https://arxiv.org/abs/2610.23683
AB - This is a continuation of our recent work~\cite{WZJ26} on asymptotic gradient regularity for nonlinear parabolic equations. In this paper, we consider bounded viscosity solutions of the degenerate or singular fully nonlinear parabolic equation u_t−|Du|^γF(D^2u)=f in Q_1, where $ γ>-1 $, $ F $ is uniformly $ (λ, Λ)$-elliptic with 0<λ≤Λ, F(0)=0, and f∈ C^0(Q_1)∩ L^∞(Q_1). Given any prescribed α∈(0,1), we prove that there exists $ \varepsilon=\varepsilon(n,α)>0 $ such that |γ|+(Λ/λ−1) ≤ ε Longrightarrow u ∈ C_loc^1+α,(1+α)/2(Q_1). In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $1$ as $ (γ, \fracΛλ) \to (0,1) $. This provides a fully nonlinear counterpart to the asymptotic gradient regularity result established in~\cite{WZJ26}.
ER -