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arXiv · 2208.11746

Nonlocal Bounded Variations with Applications

Harbir Antil·Hugo Díaz·Tian Jing·Armin Schikorra
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Abstract

Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ($BV$)-type spaces. Two different natural fractional analogs of classical $BV$ are considered: $BV^α$, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and $bv^α$, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics - this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter $bv^α$ actually corresponds to the Gagliardo-Slobodeckij space $W^{α,1}$. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel pre-dual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.

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Harbir Antil, Hugo Díaz, Tian Jing, Armin Schikorra. 2022-08-24. Nonlocal Bounded Variations with Applications. https://arxiv.org/abs/2208.11746

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