Coarea smoothing with fixed boundary values and area control

Let $(R,k)$ be a compact smooth Riemannian three-region with boundary $\mathrm{\partial }R=B_{-}\bigsqcup B_{+}$ consisting of two smooth boundary components. Let $u$ be a real Lipschitz function, equal to constants $c_{-}$ and $c_{+}$ on neighborhoods of $B_{-}$ and $B_{+}$, where $c_{-}\le \inf I<\sup I\le c_{+}$ for a nonempty open interval $I$. Suppose ${\mathcal{H}}_k^2(\{u=t\})\le A<\mathrm{\infty }$ for almost every $t\in I$, with $A\ge 0$ and normalized metric Hausdorff measure. Then there is a sequence $u_j\in C^{\mathrm{\infty }}(R)$ converging to $u$ in $C^0\cap W^{1,1}(R,k)$, with the exact same constants on fixed boundary neighborhoods. For every $ϵ>0$, some $u_j$ has a regular value $t_j\in I$ with interior level $D_{ϵ}=u_j^{-1}(t_j)$ separating the boundary components with $\mathrm{\mid }{D}_{ϵ}{\mathrm{\mid }}_k\le A+ϵ$. The set ${\mathrm{\Omega }}_{ϵ}=\{u_j\le t_j\}$ is a smooth compact region with $\mathrm{\partial }{\mathrm{\Omega }}_{ϵ}=B_{-}\bigsqcup D_{ϵ}$. The surface may be disconnected; if the boundary components can be joined by a path in $R$, it is nonempty. An endpoint-map area bound is a separate geometric input and is not asserted here.

The area hypothesis is an input on the original Lipschitz levels. It may be supplied by a correctly justified endpoint construction, but this result requires no endpoint map, multiplicity-one assertion or regularity of the original cut. The independent coarea theorem is recorded in its helper; relative C0 and W11 approximation is proved locally by finite chart mollification rather than imported from a theorem asserting only Lipschitz-controlled uniform approximation.