Vol. 13, No. 2, 2020

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Optimal constants for a nonlocal approximation of Sobolev norms and total variation

Clara Antonucci, Massimo Gobbino, Matteo Migliorini and Nicola Picenni

Vol. 13 (2020), No. 2, 595–625
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.

Keywords
Gamma-convergence, Sobolev spaces, bounded-variation functions, monotone rearrangement, nonlocal functional, nonconvex functional
Mathematical Subject Classification 2010
Primary: 26B30, 46E35
Milestones
Received: 27 May 2018
Revised: 23 December 2018
Accepted: 7 March 2019
Published: 19 March 2020
Authors
Clara Antonucci
Scuola Normale Superiore
Pisa
Italy
Massimo Gobbino
Università degli Studi di Pisa
Pisa
Italy
Matteo Migliorini
Scuola Normale Superiore
Pisa
Italy
Nicola Picenni
Scuola Normale Superiore
Pisa
Italy