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Boundary Regularity for Fully Nonlinear Singular Elliptic Equations
Abstract
We study boundary regularity for the fully nonlinear singular elliptic equation of the form |Du|γF(D2u)=f, where −1<γ<0. First, we prove pointwise boundary C1,α estimates with pointwise C1,α Dirichlet data, which weakens the C2 domain smoothness requirement from Birindelli and Demengel~\cite{BD10}. The proof uses compactness and perturbation, without flattening the boundary. Second, we extend the W2,δ estimates of Li and Li~\cite{LiLi17} from balls to C1,α domains with C1,α Dirichlet data{, for 0<δ≤δ0<1. Here δ0 is an exponent in the interior W2,δ0 estimate}. The proof combines pointwise boundary C1,α estimates, interior W2,δ0 estimates, and a Whitney decomposition.
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BibTeX
@misc{dong2026boundary,
title = {{Boundary Regularity for Fully Nonlinear Singular Elliptic Equations}},
author = {Jiqi Dong and Xuemei Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34238},
eprint = {2609.34238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34238},
}
RIS
TY - UNPB
TI - Boundary Regularity for Fully Nonlinear Singular Elliptic Equations
AU - Jiqi Dong
AU - Xuemei Li
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34238
UR - https://arxiv.org/abs/2609.34238
AB - We study boundary regularity for the fully nonlinear singular elliptic equation of the form |Du|^γF(D^2u)=f, where −1<γ<0. First, we prove pointwise boundary C^1,α estimates with pointwise C^1,α Dirichlet data, which weakens the C^2 domain smoothness requirement from Birindelli and Demengel~\cite{BD10}. The proof uses compactness and perturbation, without flattening the boundary. Second, we extend the W^2,δ estimates of Li and Li~\cite{LiLi17} from balls to C^1,α domains with C^1,α Dirichlet data{, for 0<δ≤δ_0<1. Here δ_0 is an exponent in the interior W^2,δ_0 estimate}. The proof combines pointwise boundary C^1,α estimates, interior W^2,δ_0 estimates, and a Whitney decomposition.
ER -