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2020 Optimal constants for a nonlocal approximation of Sobolev norms and total variation
Clara Antonucci, Massimo Gobbino, Matteo Migliorini, Nicola Picenni
Anal. PDE 13(2): 595-625 (2020). DOI: 10.2140/apde.2020.13.595

Abstract

We consider the family of nonlocal and nonconvex 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 1.

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 1, and then to a finite combinatorial rearrangement inequality.

Citation

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Clara Antonucci. Massimo Gobbino. Matteo Migliorini. Nicola Picenni. "Optimal constants for a nonlocal approximation of Sobolev norms and total variation." Anal. PDE 13 (2) 595 - 625, 2020. https://doi.org/10.2140/apde.2020.13.595

Information

Received: 27 May 2018; Revised: 23 December 2018; Accepted: 7 March 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07181511
MathSciNet: MR4078237
Digital Object Identifier: 10.2140/apde.2020.13.595

Subjects:
Primary: 26B30 , 46E35

Keywords: bounded-variation functions , Gamma-convergence , monotone rearrangement , nonconvex functional , nonlocal functional , Sobolev Spaces

Rights: Copyright © 2020 Mathematical Sciences Publishers

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