σ-convergence of non-local, non-convex functionals in one dimension

Haïm Brezis; Hoai Minh Nguyen

doi:10.1142/S0219199719500779

σ-convergence of non-local, non-convex functionals in one dimension

Haïm Brezis, Hoai Minh Nguyen

School of Arts and Sciences, Mathematics

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

2 Scopus citations

Abstract

We study the σ-convergence of a family of non-local, non-convex functionals in Lp(I) for p ≥ 1, where I is an open interval. We show that the limit is a multiple of the W1,p(I) semi-norm to the power p when p > 1 (respectively, the BV (I) semi-norm when p = 1). In dimension one, this extends earlier results which required a monotonicity condition.

Original language	English (US)
Article number	1950077
Journal	Communications in Contemporary Mathematics
Volume	22
Issue number	7
DOIs	https://doi.org/10.1142/S0219199719500779
State	Published - Nov 1 2020

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Non-local
Sobolev norms
non-convex
pointwise convergence
σ-convergence

Access to Document

10.1142/S0219199719500779

Cite this

@article{2383cab9cdbb416c96ff59bedf6d6003,

title = "σ-convergence of non-local, non-convex functionals in one dimension",

abstract = "We study the σ-convergence of a family of non-local, non-convex functionals in Lp(I) for p ≥ 1, where I is an open interval. We show that the limit is a multiple of the W1,p(I) semi-norm to the power p when p > 1 (respectively, the BV (I) semi-norm when p = 1). In dimension one, this extends earlier results which required a monotonicity condition.",

keywords = "Non-local, Sobolev norms, non-convex, pointwise convergence, σ-convergence",

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

note = "Publisher Copyright: {\textcopyright} 2020 World Scientific Publishing Company.",

year = "2020",

month = nov,

day = "1",

doi = "10.1142/S0219199719500779",

language = "English (US)",

volume = "22",

journal = "Communications in Contemporary Mathematics",

issn = "0219-1997",

publisher = "World Scientific",

number = "7",

}

TY - JOUR

T1 - σ-convergence of non-local, non-convex functionals in one dimension

AU - Brezis, Haïm

AU - Nguyen, Hoai Minh

PY - 2020/11/1

Y1 - 2020/11/1

N2 - We study the σ-convergence of a family of non-local, non-convex functionals in Lp(I) for p ≥ 1, where I is an open interval. We show that the limit is a multiple of the W1,p(I) semi-norm to the power p when p > 1 (respectively, the BV (I) semi-norm when p = 1). In dimension one, this extends earlier results which required a monotonicity condition.

AB - We study the σ-convergence of a family of non-local, non-convex functionals in Lp(I) for p ≥ 1, where I is an open interval. We show that the limit is a multiple of the W1,p(I) semi-norm to the power p when p > 1 (respectively, the BV (I) semi-norm when p = 1). In dimension one, this extends earlier results which required a monotonicity condition.

KW - Non-local

KW - Sobolev norms

KW - non-convex

KW - pointwise convergence

KW - σ-convergence

UR - http://www.scopus.com/inward/record.url?scp=85076799227&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076799227&partnerID=8YFLogxK

U2 - 10.1142/S0219199719500779

DO - 10.1142/S0219199719500779

M3 - Article

AN - SCOPUS:85076799227

SN - 0219-1997

VL - 22

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 7

M1 - 1950077

ER -

σ-convergence of non-local, non-convex functionals in one dimension

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this