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 -