发表机构
Seoul National University（首尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将Adimurthi–Phuc的近自然Calderón–Zygmund理论推广到矩阵退化情形，针对$A_p$加权拟线性椭圆方程的很弱解，利用比较估计、更高可积性等加权分析技术，建立了自然指数附近的局部梯度估计。
AI 中文摘要
我们建立了矩阵加权拟线性方程很弱解在自然指数附近的局部Calderón–Zygmund估计，方程形式为$- \mathrm{div\\\\,} A_{\mathbb{M}}(x,Du) = - \mathrm{div\\\\,} A_{\mathbb{M}}(x,\mathbf{f})$，其中$A_{\mathbb{M}}(x,ξ)=\mathbb{M}(x)A(x,\mathbb{M}(x)ξ)$，$A$具有$p$增长性和强单调性，$\mathbb{M}$是可测正定矩阵场。我们假设$\mathbb{M}$的条件数有界，且$\omega:=|\mathbb{M}|^p\in A_p$，不对$\mathbb{M}$施加一致上下界约束。这将Adimurthi–Phuc提出的近自然Calderón–Zygmund理论推广到了矩阵退化情形：存在$δ_0>0$，使得每个很弱解$u \in W^{p-δ_{0}}_{\omega, \mathrm{loc}}$满足：当$p-δ_0\leγ\le p+δ_0$时，若$\mathbf{f}\in L^γ_{\omega,\mathrm{loc}}$则$Du\in L^γ_{\omega,\mathrm{loc}}$。证明需要处理自然指数以下缺乏能量估计的问题，以及在加权框架下使用Lipschitz截断，我们通过比较估计、更高可积性和加权分析技术实现了这一目标。
英文摘要
We establish local Calderón--Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations\[ - \mathrm{div\,} A_{\mathbb{M}}(x,Du) = - \mathrm{div\,} A_{\mathbb{M}}(x,\mathbf{f}), \qquad A_{\mathbb{M}}(x,ξ)=\mathbb{M}(x)A(x,\mathbb{M}(x)ξ), \] where $A$ has $p$-growth and strong monotonicity, and $\mathbb{M}$ is a measurable positive-definite matrix field. We assume that $\mathbb{M}$ has bounded condition number and that \(ω:=|\mathbb{M}|^p\in A_p\), without imposing uniform upper or lower bounds on $\mathbb{M}$. 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 $u \in W^{p-δ_{0}}_{ω, \mathrm{loc}}$ satisfies \[ \mathbf{f}\in L^γ_{ω,\mathrm{loc}} \Longrightarrow Du\in L^γ_{ω,\mathrm{loc}} \] for $p-δ_0\leγ\le p+δ_0$. 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.
Comments27 pages
