A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I

Abstract

We continue the study of the space $B{V}^{\alpha }({\mathbb{R}}^n)$ of functions with bounded fractional variation in ${\mathbb{R}}^n$ of order $\alpha \in (0,1)$ introduced in arXiv:1809.08575, by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of our previous work, we prove that the fractional $\alpha$-variation converges to the standard De Giorgi's variation both pointwise and in the $\Gamma$-limit sense as $\alpha \to 1^{-}$. We also prove that the fractional $\beta$-variation converges to the fractional $\alpha$-variation both pointwise and in the $\Gamma$-limit sense as $\beta \to {\alpha }^{-}$ for any given $\alpha \in (0,1)$.

61 pages

References in corpus (5)

• Gamma-convergence of nonlocal perimeter functionals
• A surprising formula for Sobolev norms
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• Regularity for a fractional p-Laplace equation
• Bourgain-Brezis-Mironescu Domains

Cited by in corpus (12)

• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• The Cheeger problem in abstract measure spaces
• The fractional variation and the precise representative of $B{V}^{\alpha ,p}$ functions
• On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative
• Failure of the local chain rule for the fractional variation
• Nonlocal-interaction vortices
• On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions
• Extending linear growth functionals to functions of bounded fractional variation
• Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula
• $H$-compactness for nonlocal linear operators in fractional divergence form
• Nonlocal Lagrange multipliers and transport densities