Gradient Estimates Near the Natural Exponent for Very Weak Solutions to Ap-Weighted Quasilinear Elliptic Equations
Abstract
We establish local Calderón--Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations− div A𝕄(x,Du) = − div A𝕄(x,f), A𝕄(x,ξ)=𝕄(x)A(x,𝕄(x)ξ), where A has p-growth and strong monotonicity, and 𝕄 is a measurable positive-definite matrix field. We assume that 𝕄 has bounded condition number and that ω:=|𝕄|p∈ Ap, without imposing uniform upper or lower bounds on 𝕄. This extends the near-natural Calderón--Zygmund theory developed by Adimurthi--Phuc \cite{AP15} to the matrix-degenerate setting: There exists δ0>0 such that every very weak solution u ∈ Wp−δ0ω, loc satisfies f∈ Lγω,loc Longrightarrow Du∈ Lγω,loc for p−δ0≤γ≤ p+δ0. The proof requires handling the lack of energy estimates below the natural exponent and the use of Lipschitz truncation in the weighted setting. We achieve this through comparison estimates, higher integrability, and weighted analysis techniques.
Related
- Solutions stable outside a compact set for anisotropic quasilinear equations with negative exponent or exponential nonlinearity
- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
- Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
- Schauder estimates for quasilinear parabolic equations with general Orlicz growth
- Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations
BibTeX
@misc{byun2026gradient,
title = {{Gradient Estimates Near the Natural Exponent for Very Weak Solutions to $A_p$-Weighted Quasilinear Elliptic Equations}},
author = {Sun-Sig Byun and Minkyu Lim},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30155},
eprint = {2608.30155},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30155},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Gradient Estimates Near the Natural Exponent for Very Weak Solutions to A_p-Weighted Quasilinear Elliptic Equations
AU - Sun-Sig Byun
AU - Minkyu Lim
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30155
UR - https://arxiv.org/abs/2608.30155
AB - We establish local Calderón--Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations− div A_𝕄(x,Du) = − div A_𝕄(x,f), A_𝕄(x,ξ)=𝕄(x)A(x,𝕄(x)ξ), where A has p-growth and strong monotonicity, and 𝕄 is a measurable positive-definite matrix field. We assume that 𝕄 has bounded condition number and that ω:=|𝕄|^p∈ A_p, without imposing uniform upper or lower bounds on 𝕄. This extends the near-natural Calderón--Zygmund theory developed by Adimurthi--Phuc \cite{AP15} to the matrix-degenerate setting: There exists δ_0>0 such that every very weak solution u ∈ W^p−δ_0_ω, loc satisfies f∈ L^γ_ω,loc Longrightarrow Du∈ L^γ_ω,loc for p−δ_0≤γ≤ p+δ_0. The proof requires handling the lack of energy estimates below the natural exponent and the use of Lipschitz truncation in the weighted setting. We achieve this through comparison estimates, higher integrability, and weighted analysis techniques.
ER -