Suppose $S$ is a smooth surface without boundary and $f:S\to\mathbb R$ is continuous. Then, we can define the total variation of $f$ as: $$ \mathrm{TV}[f]:=\sup_{\|\phi\|_\infty\leq1} \int_S f(x)\nabla\cdot\phi(x)\,dA(x). $$ Here the variable $\phi$ is a smooth vector field whose norm is pointwise bounded by $1$. When $f$ is differentiable, this expression is the same as $\int_S \|\nabla f(x)\|_2\,dA(x)$.
Now, suppose $\overline S=\cup_{i=1}^k \overline S_i$ is the union of $k$ smooth surfaces that join along potentially sharp crease curves. It still makes sense to define a function $f$ to be continuous on $\overline S$, and $f$ can even be differentiable away from $\cup_i \partial S_i$.
Is there a definition of total variation of a function $f$ on the creased surface $\overline S$ that agrees with the formula above? In particular, we no longer can use $\nabla\cdot\phi$ on the creases since it's not clear what it means to have divergence (I think).
Ideally I'd like the formula to still work when the $\overline S$ is smooth without creases, and if we take a limiting sequence of smoothed-out versions of $\overline S$ (whatever that means) and apply the smooth formula, then we get our general formula in the limit.
Furthermore, if $f$ is smooth on each $\overline S_i$ but potentially discontinuous along the edges between the patches, I'm hoping for a formula similar to Gauss-Bonnet with integrals along the creases: $$ \mathrm{TV}_{\overline S}[f] = \sum_i \int_{\overline S_i} \|\nabla f(x)\|\,dA(x) + \sum_i \int_{\partial \overline S_i} [\mathrm{something}]\,d\ell. $$ This seems possible via integration by parts.
Any pointers to papers or formulations would be much appreciated!