具有 Dini 系数的椭圆方程几何集
Geometric sets of elliptic equations with Dini coefficients
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中文总结 AI 辅助
本文研究具有 Dini 系数的散度型椭圆方程，建立加倍指标的几乎单调性，获得临界集和节点集的体积估计，并证明切线映射的定量唯一性及 Minkowski 型估计。
中文摘要 AI 辅助
我们研究具有 Dini 系数的散度型椭圆方程，扩展了 Lipschitz 系数框架（参见 \cite{NV}）和 Hölder 系数框架（参见 \cite{HJ1}）。我们首先在仅连续系数下建立了加倍指标的几乎单调性性质。这导致了对临界集的弱体积估计和对节点集的均匀体积估计。Dini 假设在此是必不可少的，因为我们构造了一个具有连续系数和有界加倍指标但其节点集具有无限测度的例子。我们还证明了子水平集的多项式增长估计。使用另一种方法，我们建立了切线映射的定量唯一性和锥分裂原理，从而得到临界集的显式 Minkowski 型估计。最后，结合 Kenig--Zhao \cite{KZ4} 的结果，我们的估计给出了边界临界集的测度界。
英文摘要
We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig--Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
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