带有两个调节系数的广义双相问题的梯度估计
Gradient estimates for generalized double phase problems with two modulating coefficients
- Kyungpook National University（庆北国立大学）
- Kongju National University（公州国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对带两个调节系数的广义双相问题建立Calderón-Zygmund梯度估计，扩展了相关经典研究的结果，验证了局部解梯度的可积性继承性质。
AI中文摘要：
我们针对散度形式的非均匀椭圆方程的解建立了Calderón-Zygmund估计，该方程基于广义双相结构Ψ(x,z)=a(x)G(|z|)+b(x)H(|z|)建模，其中G和H是Young函数，a、b是非负的Hölder连续系数，满足自然非退化条件a(·)+b(·)≥μ>0。在对G、H的自然假设以及a、b的Hölder正则性条件下，我们证明了任意局部解的梯度继承了与数据相同的可积性；更准确地说，若Ψ(·,F)∈L^Θ_loc，则对所有Θ∈N，都有Ψ(·,Du)∈L^Θ_loc。我们的结果将Baasandorj-Byun-Oh（发表于《J. Funct. Anal.》第279卷第7期，2020年）的研究结果从经典广义双相结构G+a(x)H扩展到了双调节系数的情形，并通过在双调节系数框架内的边界情形下为广义双相泛函建立Calderón-Zygmund估计，扩展了Kim-Kim-Oh（发表于《Nonlinear Differ. Equ. Appl.》第33卷，2026年）的梯度估计成果。
英文摘要：
We prove local Calderón-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density $a(x)G(|z|)+b(x)H(|z|)$, where $G$ and $H$ are Young functions and $a(\cdot)$, $b(\cdot)$ are nonnegative coefficients, Hölder continuous with exponents $α$ and $β$. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to $q/p\le1+\min\{α,β\}/n$ for $G(t)=t^p$ and $H(t)=t^q$, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.
