Ginzburg-Landau Vortices

188 Pages·1994·14.647 MB·English

by  Fabrice Bethuel, Haïm Brezis, Frédéric Hélein (auth.)

Progress in Nonlinear Differential Equations
and TheirApplications
Volume 13
Editor
HaimBrezis
Universit~ PierreetMarieCurie
Paris
and
Rutgers University
NewBrunswick, N.J.
EditorialBoard
A. Bahri, Rutgers University, New Brunswick
John Ball, Heriot-WattUniversity,Edinburgh
LuisCafarelli, InstituteforAdvanced Study,Princeton
MichaelCrandall, UniversityofCalifornia, SantaBarbara
MarianoGiaquinta, UniversityofAorence
DavidKinderlehrer, Carnegie-MellonUniversity,Pittsburgh
Robert Kohn, New YorkUniversity
P. L. Lions, UniversityofParisIX
Louis Nirenberg, New YorkUniversity
LambertusPeletier,UniversityofLeiden
PaulRabinowitz, UniversityofWisconsin,Madison
Fabrice Bethuel
HrumBrezis
Fr6d6ric H6lein
Ginzburg-Landau Vortices
Springer Science+Business Media, LLC
Fabrice Bethuel Halm Brezis
Laboratoire d'Analyse Num~rique Analyse Num~rique
Universit~ Paris-Sud Universit6 Pierre et Marie Curie
91405 Orsay Cedex 4, place Jussieu
France 75252 Paris Cedex 05, France
and
Fred~ric H~lein
Department of Mathematics
CMLA, ENS-Cachan
Rutgers University
94235 Cachan Cedex
New Brunswick, NJ 08903
France
Library of Congress Cataloging-in-Publication Data
Bethuel, Fabrice, 1%3-
Ginzburg-Landau vortices I Fabrice Bethuel, Halm Brezis, Frederic
Helein.
p. cm. --(Progress in nonIinear differential equations and
their applications ; v. 13)
Included bibliographical references and index.
ISBN 978-0-8176-3723-1 ISBN 978-1-4612-0287-5 (eBook)
DOI 10.1007/978-1-4612-0287-5
1. Singularities (Mathematics) 2. Mathematical physics.
3. Superconductors--Mathematics. 4. Superfluidity--Mathematics.
5. Differential equations, Nonlinear--Numberical solutions.
I. Brezis, H. (Haim) n. Helein, Frederic, 1%3- m. Title.
IV. Series.
QC20.7.S54B48 1994 94-2026
530.1 '55353--dc20 CIP
m®
Printed on acid-free paper
© Springer Science+Business Media New York 1994 a{Jl)
Originally published by Birkhauser Boston in 1994
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ISBN 978-0-8176-3723-1
Typeset by the Authors in AMSTEX.
9876.5432
TABLE OF CONTENTS
Introduction ix
I. Energy estimates for SI-valued maps
1. An auxiliary linear problem 1
2. Variants ofTheorem 1.1 6
3. 51-valued harmonic maps with prescribed isolated
singularities. The canonical harmonic map 10
4. Shrinking holes. Renormalized energy 16
II. A lower bound for the energy of SI-valued maps
on perforated domains 31
III. Some basic estimates for Ue
= =
1. Estimates when G BR and g(x) x/lxl 42
2. An upper bound for Ee(ue) 44
3. An upper bound for fz fG(luel2 - 1)2 45
4. Iuel 2: 1/2 on "good discs" 46
IV. Towards locating the singularities:
bad discs and good discs
1. A covering argument 48
2. Modifying the bad discs 49
V. An upper bound for the energy of away from
Ue
the singularities
1. A lower bound for the energy of near 53
Ue aj
2. ProofofTheorem V.1 54
CONTENTS
VI. U converges: U* is born!
En
1. ProofofTheorem VI.1 58
2. Further properties of u* : singularities have degree
one and they are not on the boundary 60
VII. u* coincides with THE canonical harmonic map
having singularities (aj) 65
VIII. The configuration (aj) minimizes the renormal-
ized energy W
1. The general case 76
2. The vanishing gradient property and its various
forms 82
3. Construction ofcritical points ofthe renormalized
energy 93
4. The case G = B and g(O) =ei8 95
1
5. The case G = B and g(O) =edi8 with d ~ 2 97
1
IX. Some additional properties of
UE
1. The zeroes of 100
UE
2. The limit of {EE(UE) -1rdllog en as e - 0 101
3. fG IVluEW remains bounded as c - 0 103
4. The bad discs revisited 104
X. Non minimizing solutions ofthe Ginzburg-Landau
equation
1. Preliminary estimates; bad discs and good discs 107
IVvEI
2. Splitting 109
3. Study ofthe associated linear problems 112
4. The basic estimates: fG IVv I2 ~ Cilog cl and
E
f /VvEIP ~ C for p < 2 119
G p
5. v converges to V* 125
En
6. Properties ofv* 132
CONTENTS
XI. Open problems 137
Appendix I. Summary of the basic convergence re-
=
sults in the case where deg(g,8G) 0 142
Appendix II. Radial solutions 145
Appendix III. Quantization effects for the equation
-.!lv = v(1 - Iv12) in ]R2 147
Appendix IV. The energy of maps on perforated d~
mains revisited 148
BmLIOGRAPHY 154
INDEX 159
Acknowledgements
We are grateful to E. DeGiorgi, H. Matano, L. Nirenberg and L. Peletier for very
stimulating discussions. During the preparation of this work we have received
advice and encouragement from many people: A. Belavin, E. Brezin, N. Carl
son, S. Chanillo, B. Coleman, L.C. Evans, J. M. Ghidaglia, R. Hardt, B. Helffer,
M. Herve, R. M. Herve, D. Huse, R. Kohn, J. Lebowitz, Y. Li; F. Merle, J. Ock
endon, Y. Pomeau, T. Riviere, J. Rubinstein, I. Shafrir, Y. Simon, J. Taylor and
F. Treves.
Part of this work was done while the first author (F.B.) and the third author
(F.H.) were visiting Rutgers University. They thank the Mathematics Department
for its support and hospitality; their work was also partially supported by a Grant
ofthe French Ministry ofResearch and Technology (MRT Grant 9OS0315). Part of
this work was done while thesecond author (H.B.) was visiting the Scuola Normale
Superiore ofPisa; he is grateful to the Scuola for its invitation. Wealso thank Lisa
Magretto and Barbara Miller for their enthusiastic and competent typing of the
manuscript.
INTRODUCTION
The original motivation of this study comes from the following questions
that were mentioned to one ofus by H. Matano.
Let
G =B ={x =(X1lX2) E22; x~ +x~ =Ixl2< 1}.
1
Consider the Ginzburg-Landau functional
L L
E~(u) ~ 2 4~2 2
(1) = IVul + (lu1 _1)2
which is defined for maps u E H1(G;C) also identified with Hl(G;R2).
Fix the boundary condition
9(X) =X on 8G
and set
=
H; = {u E H1(G;C); u 9 on 8G}.
It is easy to see that
(2)
is achieved by some u~ that is smooth and satisfies the Euler equation
-~u~ u~(1 _lu~12)
= :2 in G,
(3)
{
u~ =9 on aGo
Themaximum principleeasily implies (seee.g., F. Bethuel, H. Brezisand F.
Helein (2]) that any solution u~ of (3) satisfies lu~1 ~ 1in G. In particular,
a subsequence (u~,.) converges in the w* - LOO(G) topology to a limit u*.
Clearly, lu*(x)1 ~ 1 a.e. It is very easy to prove (see Chapter III) that
L
(lu~12 ~ 2
(4) - 1)2 Ce 1logel
and thus lu~"(x)1 -. 1 a.e. This suggests that lu*(x)1 = 1 a.e. However,
such a claim is not clear at all since we do not know, at this stage, that
ue" -+ u* a.e. It turns out to be true that lu*(x)1 = 1 a.e. - but we have
no simple proof. This fact is derived as a consequence ofa delicate analysis
(see Chapter VI).
x Introduction
The original questions of H. Matano were:
Question 1: Does limue(x) exist a.e.?
e.....O
Question 2: What is u*? Do we have u*(x) = x/lxl?
Question 3: What can be said about the zeroesofue? Ifthey are isolated
do they have degree ±1 (in the sense of Section D(1)?
These questions have prompted us to consider a more general setting.
Let G C ]R2 be a smooth, bounded and simply connected domain in R2.
Fix a (smooth) boundary condition 9 :aG -+ 81 and consider a minimizer
Ue of problem (2) as above. Our purpose is to study the behavior ofUe as
c -+ O.
The Brouwer degree
=
(5) d deg(g,aG)
(i.e., the winding number of9 considered as a map from 8G into 81) plays
a crucial role in the asymptotic analysis ofUe•
Case d =o. This case is easy because H:(G; 81) =f: 0and thus the mini
mization problem
(6)
makessense. In fact, problem (6) has a uniquesolutionUo that is a smooth
harmonic map from G into 81, i.e.,
Moreover (see e.g., Lemma 1 in F. Bethuel, H. Brezis and F. Helein 12])
where !Po is a harmonic function (unique mod 21rZ) such that
ei'Po = 9 on aGo
WehaveprovedinF. Bethuel, H. BrezisandF. Helein 12) (seealso Appendix
I at the end of the book) that Ue -+ Uo in C1,Q(G) and in Cfoc(G) Vk; in
particular,
L
(7) IVuel2 remains bounded as c -+ O.
We have also obtained rates ofconvergence for lIue - UoII in various norms.