Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain.
It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite norm $$\lVert u \rVert_{L^1(\partial\Omega)} := \sum_i\lVert {\phi_i(u\circ g_i)}\rVert_{L^1(B_i)}$$ where $\phi_i$ partition of unity, $g_i$ is a $C^1$ diffeomorphism and $B_i$ are subsets of $\mathbb{R}^{n}$.
There are lots of definitions like this (eg. see Renardy and Rogers, Krylov, James Robinson's Infinite Dimensional Dynamical System) or with small variations.
There is also the surface integral of a function $v\colon \partial\Omega \to \mathbb{R}$: $$|u| := \int_{\partial \Omega}fdS$$ where $dS$ is the surface density. To compute this quantity, we need to use a parametrisation.
Now my question, is it always the case that $$\lVert u \rVert_{L^1(\partial\Omega)} \qquad\text{and}\qquad |u|$$ are equivalent? Does this hold for all the little variations of the $L^1$ norm in the first equation??
Now when we have a Lipschitz surface, these norms are equivalent (see Necas). BUT their definition of the $L^1$ norm is different so let us not use that result here. Edit: hmm, it seems $C^1$ domains are not a subset of Lipschitz domains.
Not, because your definition depends on the selection of the partition.
The following does not depend of the partition: $$ \|u\|_{L^1(\partial\Omega)}=\Big\|\sum_{i}\int_{B_i}\varphi_i(u\circ g_i)\,dx\,\Big\|. $$ Note that the $B_i$'s should be subsets of $\mathbb R^{n-1}$.