Families of functionals representing Sobolev norms

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

doi:10.2140/apde.2024.17.943

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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

DOI: 10.2140/apde.2024.17.943

Abstract

We obtain new characterizations of the Sobolev spaces $Ẇ^{1, p} (ℝ^{N})$ and the bounded variation space $\dot{BV} (ℝ^{N})$ . The characterizations are in terms of the functionals $ν_{γ} (E_{λ, γ ∕ p} [u])$ , where

E_{λ, γ ∕ p} [u] = {(x, y) \in ℝ^{N} \times ℝ^{N} : x \neq y, \frac{| u (x) - u (y) |}{| x - y |^{1 + γ ∕ p}} > λ}

and the measure $ν_{γ}$ is given by $d ν_{γ} (x, y) = | x - y |^{γ - N} d x d y$ . We provide characterizations which involve the $L^{p, \infty}$ -quasinorms $\sup_{λ > 0} λ ν_{γ} {(E_{λ, γ ∕ p} [u])}^{1 ∕ p}$ and also exact formulas via corresponding limit functionals, with the limit for $λ \to \infty$ when $γ > 0$ and the limit for $λ \to 0^{+}$ 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 $γ \neq 0$ . For $p = 1$ the upper bounds for the $L^{1, \infty}$ quasinorms fail in the range $γ \in [- 1, 0)$ ; moreover, in this case the limit functionals represent the $L^{1}$ norm of the gradient for $C_{c}^{\infty}$ -functions but not for generic $Ẇ^{1, 1}$ -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 $ν_{0} (E_{λ, 0} [u])$ .

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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Vol. 17, No. 3, 2024