Riemannian Lipschitz coarea on compact interior bands

Let $(R,k)$ be a compact smooth Riemannian three-region with boundary, and let $u:R\to \mathbb{R}$ be Lipschitz for its intrinsic length distance. Suppose that $u$ is constant near each of two boundary parts whose union is $\mathrm{\partial }R$, with the two constants outside an interval $I$ on opposite sides. For every Borel interval $J\subset I$,

${\int }_{\{x\in R:u(x)\in J\}}\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }d{V}_k={\int }_J{\mathcal{H}}_k^2(\{x\in R:u(x)=t\})dt\mathrm{.}$

Here $d{V}_k$ is Riemannian volume, ${\mathrm{\nabla }}_{k}u$ is the almost-everywhere weak gradient characterized by $k({\mathrm{\nabla }}_{k}u,\cdot )=du$, and ${\mathcal{H}}_k^2$ is normalized two-dimensional Hausdorff measure for the metric. A smooth $u$ satisfying the same boundary condition also satisfies the identity. More generally, for any nonnegative measurable $w$ on $R^{\circ }$,

${\int }_{R^{\circ }}w\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }d{V}_k={\int }_{\mathbb{R}}\left({\int }_{\{x\in R^{\circ }:u(x)=t\}}wd{\mathcal{H}}_k^2\right)dt\mathrm{.}$

Both sides are extended nonnegative integrals. Level area here is Hausdorff area; it is not defined to be BV perimeter.

Removing the boundary before applying scalar coarea

Lipschitz height functions, cut measure and Jacobians

Riemannian metric and riemannian manifold

The cited Riemannian coarea theorem applies to Lipschitz maps between $C^1$ Riemannian manifolds and expresses the integral of their metric Jacobian as the integral of normalized Hausdorff measures of fibers. Its weighted corollary gives the same identity for nonnegative measurable weights; its proof uses monotone convergence. The source input is used only on the boundaryless interior, while the boundary treatment and identification of the scalar Jacobian are supplied by the local dependencies.