On blow-ups of sets with finite fractional variation

TL;DR

This paper investigates the local geometric structure of sets with finite fractional variation, showing tangent sets are half-spaces oriented by fractional normals, thus extending classical geometric measure theory to fractional contexts.

Contribution

It establishes that tangent sets of fractional variation sets with finite fractional perimeter are half-spaces aligned with fractional normals, revealing new geometric properties in fractional calculus.

Findings

Tangent sets are half-spaces oriented by fractional normals.

Results extend classical geometric measure theory to fractional settings.

Provides a detailed analysis of blow-ups of fractional variation sets.

Abstract

Given $\alpha\in(0,1)$ and a set $E\subset\mathbb{R}^N$ with locally finite fractional $\alpha$-variation, we show that for $|D^\alpha\mathbf 1_E|$-a.e. $x$, every non-trivial tangent set of $E$ at $x$ with locally finite integer perimeter is a half-space oriented by the fractional inner unit normal of $E$ at $x$.