TY - JOUR
T1 - Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga-type functionals
AU - Poliakovsky, Arkady
N1 - 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
SN - 0219-1997
VL - 20
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
IS - 7
M1 - 1750096
ER -