Optimal constants for a non-local approximation of Sobolev norms and total variation

TL;DR

This paper determines explicit constants for the Gamma-convergence of non-local functionals to Sobolev norms and total variation, providing new insights into their structure and recovery families across all dimensions.

Contribution

It establishes exact constants for Gamma-convergence of non-local functionals to Sobolev norms and total variation in any dimension, and constructs smooth recovery families.

Findings

Explicit constants for Gamma-convergence in all dimensions

Existence of smooth compactly supported recovery families

Reduction to one-dimensional and combinatorial problems

Abstract

We consider the family of non-local and non-convex functionals proposed and investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series of papers of the last decade. It was known that this family of functionals Gamma-converges to a suitable multiple of the Sobolev norm or the total variation, depending on the summability exponent, but the exact constants and the structure of recovery families were still unknown, even in dimension one. We prove a Gamma-convergence result with explicit values of the constants in any space dimension. We also show the existence of recovery families consisting of smooth functions with compact support. The key point is reducing the problem first to dimension one, and then to a finite combinatorial rearrangement inequality.