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

  • Emanuele Tasso EMAIL logo

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 n and given p > 1 , we study the blow-up of functions u GSBV ( Ω ) , whose jump sets belong to an appropriate class 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 . Moreover, we are able to prove the following result which in the case of W 1 , p ( Ω ) 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 .

MSC 2010: 28A75; 26B30; 31C15

Communicated by Frank Duzaar


Acknowledgements

The author wishes to thank Prof. Gianni Dal Maso for many helpful discussions on the topic. A special thanks goes to Emanuele Caputo for his skill and patience in helping me creating the pictures.

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Received: 2019-08-07
Revised: 2019-10-20
Accepted: 2019-11-04
Published Online: 2020-01-18
Published in Print: 2022-01-01

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