平面p-调和函数的Almgren型公式
An Almgren-type formula for planar $p$-harmonic functions
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对1<p<∞的平面p-调和函数，引入通量归一化频率作为Almgren频率的非线性对应，建立其单调性并推导定量估计，明确其高维推广的障碍。
AI中文摘要：
对于满足1 < p < ∞的非常数平面p-调和函数u，结合先前结果（尤其是Aronsson对孤立临界点附近 hodograph 表示的基础分析），本文引入通量归一化频率N₊(x₀,r) = [r∫_{B_r(x₀)} |Du|ᵖ dx] / [∫_{∂B_r(x₀)} |Du|^{p-2}(u-u(x₀))² dS]。该比值是非线性情形下Almgren频率函数的自然对应，可精确还原所有齐次p-调和剖面的齐次度。本文进一步建立Almgren型单调性的规范校正版本，通过频率函数重新阐释经典平面唯一延拓性质，还推导了通量归一化频率的定量夹逼估计，并明确了阻碍该方法直接推广至高维的具体障碍。
英文摘要：
For a nonconstant planar $p$-harmonic function $u$, with $1 < p < \infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$ N_*(x_0,r) = \frac{r\displaystyle\int_{B_r(x_0)} |D u|^p\,dx} {\displaystyle\int_{\partial B_r(x_0)} |D u|^{p-2}(u-u(x_0))^2\,dS}. $$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.
