TY - CHAP
T1 - Uniqueness of lifting and beyond
AU - Brezis, Haïm
AU - Mironescu, Petru
N1 - Publisher Copyright:
© 2021, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021
Y1 - 2021
N2 - So far, wehave been concerned with the existence of a lifting φ: Ω→ R for a given u: Ω→ S1. A natural question is whether such φ is unique (mod 2 π ). More precisely, assume that we have two liftings, φ1, φ2. Then, φ1(x) - φ2(x) = 2 πk(x) for some k: Ω→ Z. Therefore, we are led to the question of finding minimal assumptions on a measurable function k: Ω→ Z implying that k must be constant. Clearly, continuity is sufficient, but, as we are going to see, constancy holds for a surprisingly large class of functions.
AB - So far, wehave been concerned with the existence of a lifting φ: Ω→ R for a given u: Ω→ S1. A natural question is whether such φ is unique (mod 2 π ). More precisely, assume that we have two liftings, φ1, φ2. Then, φ1(x) - φ2(x) = 2 πk(x) for some k: Ω→ Z. Therefore, we are led to the question of finding minimal assumptions on a measurable function k: Ω→ Z implying that k must be constant. Clearly, continuity is sufficient, but, as we are going to see, constancy holds for a surprisingly large class of functions.
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U2 - 10.1007/978-1-0716-1512-6_6
DO - 10.1007/978-1-0716-1512-6_6
M3 - Chapter
AN - SCOPUS:85122424446
T3 - Progress in Nonlinear Differential Equations and Their Application
SP - 227
EP - 252
BT - Progress in Nonlinear Differential Equations and Their Application
PB - Birkhauser
ER -