On the blow-up of GSBV functions under suitable geometric properties of the jump set
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Date
2019-06
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SISSA
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.
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Keywords
blow-up, special bounded variation, indecomposable set, jump set, perimeter, rectifiable set, Cheeger's constant, isoperimetric profile, Poincare's inequality.