*Submitted Paper*

**Inserted:** 26 jun 2019

**Last Updated:** 7 aug 2019

**Year:** 2019

**Notes:**

Preprint SISSA 18$/$2019$/$MATE

**Abstract:**

In this paper we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given $p>1$ we study the blow-up of functions $u \in GSBV$, whose jump sets belongs to an appropriate class $\mathcal{J}_p$ and whose approximate gradient is $p$-th power summable. In analogy with the theory of $p$-capacity in the context of Sobolev spaces, we prove that the blow-up of $u$ converges up to a set of Hausdorff dimension less than or equal to $n-p$. Moreover, we are able to prove the following result which in the case of $W^{1,p}(\Omega)$ functions can be stated as follows: whenever $u_k$ strongly converges to $u$, then up to subsequences, $u_k$ pointwise converges to $u$ except on a set whose Hausdorff dimension is at most $n-p$.

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