平稳积分varifold的任意有限阶可修正性
Rectifiability of every finite order for stationary integral varifolds
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中文总结 AI 辅助
本文给出平稳积分varifold支撑集任意有限阶可修正性的独立证明，通过比较光滑面积被积函数解图像获得超幂次过剩衰减，结合Santilli刻画与Whitney延拓定理。
中文摘要 AI 辅助
我们给出了Brena、De Lellis和Franceschini（arXiv:2503.00649）的一个定理的另一种证明：在$\mathbf{R}^n$的开子集中，一个平稳积分$d$维varifold的支撑集对于每个正整数$k$和每个$0<\alpha<1$都是$(\mathscr{H}^d, d)$类$(k, \alpha)$可修正的，特别是$\mathscr{C}^{\infty}$类。我们的证明独立于他们的证明，并采用不同的方法。它在每个尺度上且在$\mathscr{H}^d$几乎处处点上，将varifold与一个光滑面积被积函数的Euler-Lagrange系统的解的图像进行比较，从而导出一个比尺度的任何幂次都更快的过剩衰减估计；结论随后由Santilli的高阶可修正性刻画以及Whitney延拓定理得出。
英文摘要
We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral $d$ dimensional varifold in an open subset of $\mathbf{R}^n$ is $(\mathscr{H}^d, d)$ rectifiable of class $(k, α)$ for every positive integer $k$ and every $0 < α< 1$, and in particular of class $\mathscr{C}^{\infty}$. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at $\mathscr{H}^d$ almost every point, with the graph of a solution of the Euler--Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli's characterisation of higher order rectifiability together with Whitney's extension theorem.
