Legendrian cycles and Reilly-type variational formulae for $F_n{W}^{2,n}$-sets

We construct a natural Legendrian cycle ${\mathcal{N}}_{\mathcal{S}}$ associated with any $F_n{W}^{2,n}$-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 $\mathrm{nor}(\mathcal{S},p)$, 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.