Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.

>Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?

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**Update.** 

1. In an [answer to this post][1], it has been showed that there exist a representative $\tilde u$ of $u$ such that its graph has Hausdorff dimension equal to $N$. 

2. In a subsequent post https://mathoverflow.net/questions/327331/if-the-hausforff-dimension-of-the-graph-of-a-function-u-is-n-and-tilde-u it has been showed that a function can be zero a.e. and still its graph may have dimension strictly greater than $1$. So probably this question is better formulated in terms of *essential graph* of $u$, which possibly is equivalent to asking for the property to hold for one representative of $u$ (see Question 2 in https://mathoverflow.net/questions/327698/hausdorff-dimension-of-the-graph-of-a-bv-function-in-1-dimensional-setting)

3. In the post https://mathoverflow.net/questions/327698/hausdorff-dimension-of-the-graph-of-a-bv-function-in-1-dimensional-setting, I've asked about a simpler proof of the result in the one-dimensional setting. 


  [1]: http://tps://mathoverflow.net/a/327310/122620