Non-local, non-convex functionals converging to Sobolev norms

Haïm Brezis; Hoai Minh Nguyen

doi:10.1016/j.na.2019.111626

Non-local, non-convex functionals converging to Sobolev norms

Haïm Brezis, Hoai Minh Nguyen

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λ_δ in L^p(Ω) for p>1. We show that the limits are multiples of ∫_Ω|∇u|^p. This is a continuation of our previous work where the case p=1 was considered.

Original language	English (US)
Article number	111626
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	191
DOIs	https://doi.org/10.1016/j.na.2019.111626
State	Published - Feb 2020

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Keywords

Non-convex
Non-local
Pointwise convergence
Sobolev norms
Γ-convergence

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title = "Non-local, non-convex functionals converging to Sobolev norms",

abstract = "We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λδ in Lp(Ω) for p>1. We show that the limits are multiples of ∫Ω|∇u|p. This is a continuation of our previous work where the case p=1 was considered.",

keywords = "Non-convex, Non-local, Pointwise convergence, Sobolev norms, Γ-convergence",

author = "Ha{\"i}m Brezis and Nguyen, {Hoai Minh}",

note = "Funding Information: This work was completed during a visit of H.-M. Nguyen at Rutgers University. He thanks H. Brezis for the invitation and the Department of Mathematics for its hospitality.",

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language = "English (US)",

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T1 - Non-local, non-convex functionals converging to Sobolev norms

AU - Brezis, Haïm

AU - Nguyen, Hoai Minh

N1 - Funding Information: This work was completed during a visit of H.-M. Nguyen at Rutgers University. He thanks H. Brezis for the invitation and the Department of Mathematics for its hospitality.

PY - 2020/2

Y1 - 2020/2

N2 - We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λδ in Lp(Ω) for p>1. We show that the limits are multiples of ∫Ω|∇u|p. This is a continuation of our previous work where the case p=1 was considered.

AB - We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λδ in Lp(Ω) for p>1. We show that the limits are multiples of ∫Ω|∇u|p. This is a continuation of our previous work where the case p=1 was considered.

KW - Non-convex

KW - Non-local

KW - Pointwise convergence

KW - Sobolev norms

KW - Γ-convergence

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JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

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