Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
- $f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
- The images $\Sigma_k:=f_k(\Omega)$ are each integer rectifiable varifolds, and
- $\Sigma_k\rightarrow\Sigma_*$ as varifolds ($\Sigma_*$ is integer rectifiable).
Is it possible to make some conclusion about the relationship between $f_*$ and $\Sigma_*$? For example, can I say that the support of $\Sigma_*$ is the same as the image of $f_*$?
As of now, my only lead is the following post about Supports of Sobolev functions, which says that if $u\in W^{1,p}(\Omega,\mathbb{R})$, then there exists a representative of $u$ such that $\mathrm{graph}(u)$ is rectifiable.
