Legendrian 循环与 $F_nW^{2,n}$-集合的 Reilly 型变分公式
Legendrian cycles and Reilly-type variational formulae for $F_nW^{2,n}$-sets
AI总结：
本文为 $F_nW^{2,n}$-集合构造了自然的 Legendrian 循环，并证明了高阶平均曲率积分的 Reilly 型变分公式，将经典 Reilly 结果推广到非光滑情形。
摘要：
我们为任意 $F_nW^{2,n}$-集合 $\mathcal{S}$ 构造了一个自然的 Legendrian 循环 $\mathcal{N}_\mathcal{S}$，其中 $\mathcal{S}$ 是一个闭集，局部可描述为有限个 $(C^0\cap W^{2,n})$ 正则函数图并具有整数重数。该构造基于以下事实：在 $\mathcal{H}^n$-几乎每个两个或更多图重叠的点处，它们的法向量至多相差一个符号。作为推论，我们证明了 $\mathcal{S}$ 的高阶平均曲率积分的 Reilly 型变分公式，将经典的 Reilly 结果推广到这一非光滑情形。
英文摘要：
We construct a natural Legendrian cycle $\mathcal{N}_\mathcal{S}$ associated with any $F_nW^{2,n}$-set $\mathcal{S}$, that is, a closed set locally described as a finite union of graphs of $(C^0\cap W^{2,n})$-regular functions with integer multiplicity. The construction relies on the fact that $\mathcal{S}$ is countably $\mathcal{H}^n$-rectifiable of class $C^2$ and, at $\mathcal{H}^n$-almost every point $p\in\mathcal{S}$, the proximal unit normal bundle at $p$, denoted by $\operatorname{nor}(\mathcal{S},p)$, consists of exactly two antipodal vectors $\{u,-u\}$, even in the presence of overlapping $W^{2,n}$-graphs. As a consequence, we prove Reilly-type variational formulae for the higher-order mean curvature integrals of $\mathcal{S}$, extending the classical results of Reilly to this non-smooth setting.
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