On the blow-up of GSBV functions under suitable geometric properties of the jump set

Abstract

In this paper we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set Ω С Rn and given p > 1 we study the blow-up of functions u Є2 GSBV (Ω), whose jump sets belongs to an appropriate class Jp 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 W1,p (Ω) functions can be stated as follows: whenever uk strongly converges to u, then up to subsequences, uk pointwise converges to u except on a set whose Hausdorff dimension is at most n - p.