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Families of functionals representing Sobolev norms

Haïm Brezis, Andreas Seeger, Jean Van Schaftingen and Po-Lam Yung

Vol. 17 (2024), No. 3, 943–979
Abstract

We obtain new characterizations of the Sobolev spaces 1,p(N) and the bounded variation space BV ˙(N). The characterizations are in terms of the functionals νγ(Eλ,γp[u]), where

Eλ,γp[u] ={(x,y) N × N : xy, |u(x) u(y)| |x y|1+γp > λ}

and the measure νγ is given by d νγ(x,y) = |x y|γN d xd y. We provide characterizations which involve the Lp,-quasinorms sup λ>0 λνγ(Eλ,γp[u])1p and also exact formulas via corresponding limit functionals, with the limit for λ when γ > 0 and the limit for λ 0+ when γ < 0. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For p > 1 the characterizations hold for all γ0. For p = 1 the upper bounds for the L1, quasinorms fail in the range γ [1,0); moreover, in this case the limit functionals represent the L1 norm of the gradient for Cc-functions but not for generic 1,1-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension γ + 1. For γ = 0 the characterizations of Sobolev spaces fail; however, we obtain a new formula for the Lipschitz norm via the expressions ν0(Eλ,0[u]).

Keywords
Sobolev norms, nonconvex functionals, nonlocal functionals, Marcinkiewicz spaces, Cantor sets and functions
Mathematical Subject Classification
Primary: 26D10, 26A33
Secondary: 35A23, 42B25, 42B35, 46E30, 46E35
Milestones
Received: 24 September 2021
Revised: 5 July 2022
Accepted: 11 August 2022
Published: 24 April 2024
Authors
Haïm Brezis
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Departments of Mathematics and Computer Science
Technion, Israel Institute of Technology
Haifa
Israel
Laboratoire Jacques-Louis Lions
Sorbonne Universités, UPMC Université Paris-6
Paris
France
Andreas Seeger
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Jean Van Schaftingen
Institut de Recherche en Mathématique et Physique
Université catholique de Louvain
Louvain-la-Neuve
Belgium
Po-Lam Yung
Mathematical Sciences Institute
Australian National University
Canberra
Australia

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