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

Abstract

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 ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} and given p > 1 {p>1} , we study the blow-up of functions u ∈ GSBV ⁢ ( Ω ) {u\in\mathrm{GSBV}(\Omega)} , whose jump sets belong to an appropriate class 𝒥 p {\mathcal{J}_{p}} and whose approximate gradients are 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 {n-p} . Moreover, we are able to prove the following result which in the case of W 1 , p ⁢ ( Ω ) {W^{1,p}(\Omega)} functions can be stated as follows: whenever u k {u_{k}} strongly converges to u, then, up to subsequences, u k {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most n - p {n-p} .