TY - JOUR

T1 - Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga-type functionals

AU - Poliakovsky, Arkady

N1 - Funding Information:
The research was supported by the Israel Science Foundation (Grant No. 999/13). I thank Itai Shafrir for some interesting discussions and Petru Mironescu for very helpful suggestions that helped me to improve an earlier version of the manuscript.
Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - Motivated by the formula, due to Bourgain, Brezis and Mironescu, (equation presented) that characterizes the functions in Lq that belong to W1,q (for q > 1) and BV (for q = 1), respectively, we study what happens when one replaces the denominator in the expression above by |x - y|. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV function. We further identify the function space relevant to the study of these functionals, the space BVq, as the Besov space Bq,∞ 1/q. We show, among other things, that BVq(Ω) contains both the spaces BV (Ω) ∩ L∞(O) and W1/q,q(Ω). We also present applications to the study of singular perturbation problems of Aviles-Giga type.

AB - Motivated by the formula, due to Bourgain, Brezis and Mironescu, (equation presented) that characterizes the functions in Lq that belong to W1,q (for q > 1) and BV (for q = 1), respectively, we study what happens when one replaces the denominator in the expression above by |x - y|. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV function. We further identify the function space relevant to the study of these functionals, the space BVq, as the Besov space Bq,∞ 1/q. We show, among other things, that BVq(Ω) contains both the spaces BV (Ω) ∩ L∞(O) and W1/q,q(Ω). We also present applications to the study of singular perturbation problems of Aviles-Giga type.

KW - Besov spaces

KW - functions of bounded variation

KW - singular perturbation problems

UR - http://www.scopus.com/inward/record.url?scp=85038364902&partnerID=8YFLogxK

U2 - 10.1142/S0219199717500961

DO - 10.1142/S0219199717500961

M3 - Article

AN - SCOPUS:85038364902

VL - 20

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 7

M1 - 1750096

ER -