调和函数节点体积的一个尖锐下界
Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions
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中文总结 AI 辅助
本文证明调和函数在球内节点体积的下界与加倍指数呈线性关系，并确认了相关民间猜想。
中文摘要 AI 辅助
设$u$是$B_4\subset\mathbb{R}^n$（$n\geq3$）中的非零实值调和函数，且$u(0)=0$。我们证明$$\mathcal{H}^{n-1}\bigl(\{u(x)=0\}\cap B_2\bigr)\ge C_n\mathcal{N},$$其中$C_n$是仅依赖于$n$的正常数，$\mathcal{N}$是由$$\mathcal{N}=\log_2\frac{\sup_{B_1}|u|}{\sup_{B_{\frac{1}{2}}}|u|}$$定义的加倍指数。对$\mathcal{N}$的线性依赖是最优的。该估计证实了关于调和函数节点体积的一个民间猜想。
英文摘要
Let $B=B(p, 1)\subset\mathbb{R}^n$ be a unit ball and $n\ge 3$. We prove that there are positive constants $c$ and $C$, depending only on $n$, such that every non-zero real-valued harmonic function $u: 4B\to \mathbb{R}$ with $u(p)=0$ satisfies $$ \mathcal{H}^{n-1}\bigl(\{u=0\}\cap 2B\bigr)\ge C\mathcal{N}, $$ and $$ \mathcal{H}^n\bigl(\{u>0\}\cap {\frac 12}B\bigr) \ge c\bigl(\log(1+\mathcal{N})\bigr)^{1-n}, $$ where $\mathcal{N}$ is the doubling index defined by $$\mathcal{N}=\log_2\frac{\sup_{B}|u|}{\sup_{\frac{1}{2}B}|u|}.$$ The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on $\mathcal{N}$ is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.
发表机构
- School of Sciences, Great Bay University（广东理工学院理学院）
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