Topics on Concentration Phenomena and Problems with Multiple Scales

325 Pages·2006·3.442 MB·English

by  Andrea Braides, Stella Gelli Maria (auth.), Andrea Braides, Valeria Chiadò Piat (eds.)

Lecture Notes of 2
the Unione Matematica Italiana
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Andrea Braides Valeria Chiadò Piat (Eds.)

Topics
on Concentration
Phenomena and Problems
with Multiple Scales
ABC
Editors
Andrea Braides Valeria Chiadò Piat
Dipartimento di Matematica Dipartimento di Matematica
Università di Roma "Tor Vergata" Politecnico di Torino
via della Ricerca Scientifica 1 Corso Duca degli Abruzzi 24
00133 Roma 10129 Torino
Italy Italy
e-mail: [email protected] e-mail: [email protected]
LibraryofCongressControlNumber: 2006928616
MathematicsSubjectClassification(2000):49-06, 49J40, 49J45, 49K 20, 49J55, 28A 80,
74Q 05, 35B27, 35B25, 35B 40
ISSNprintedition:1862-9113
ISSNelectronicedition:1862-9121
ISBN-10 3-540-36241-x SpringerBerlinHeidelbergNewYork
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Preface
The research group ‘Homogenization Techniques and Asymptotic Methods
for Problems with Multiple Scales’, co-ordinated by Valeria Chiado` Piat and
funded by INdAM-GNAMPA (Istituto Nazionale di Alta Matematica-Gruppo
Nazionale per l’Analisi Matematica, la Probabilita` e le loro Applicazioni), op-
eratedfrom2001to2005,involvinginitsactivitiesanumberofyoungItalian
mathematicians, mainly interested in problems in the Calculus of Variations
and Partial Differential Equations. One of the initiatives of that group has
been the organization of a number of schools. Those in the years 2001-2003,
whose lecture notes are gathered in this book, had been devoted to problems
with oscillations and concentrations, while the schools in the years 2004-2005
covered a range of topics of Applied Mathematics.
The first school in Turin, 17–21 September 2001, bearing the name of the
research group and devoted to problems with multiple scales, was partially
disrupted by the events of September 11 of that year, one speaker, Gilles
Francfort, finding himself grounded in Los Angeles, and two other speakers,
Andrey Piatnitski and Gregory Chechkin, slowed down in their car trip to
Italy by the tightening of the borders around the European Community. The
two remaining speakers managed however to enlarge their courses to cover
some extra material, encouraged by the receptive audience. The course of
AndreaBraideswasdevotedtothedescriptionofthebehaviourofvariational
problemsonlatticesasthelatticespacingtendstozero,andthevariousmulti-
scale behaviours that may be obtained from this process; that of Anneliese
Defranceschi to energies with competing bulk and surface interactions. The
extra lectures are not included in these notes, but some of them constitute
part of the material in the book ‘Γ-convergence for Beginners’ (Oxford U.P.,
2002)byBraides.ThecourseofFrancfort,onH-measures,waslaterrecovered
ina‘PartII’oftheSchoolheldatIACinRome,December3–5,2001,together
with a contribution of Roberto Peirone on homogenization on fractals. Here
we also include the text of the two courses of Piatnitski and Chechkin, while
thelecturenotesofthecoursebyFrancforthaveappearedasachapterofthe
book ‘Variational Problems in Materials Science’, Birkha¨user, 2006.
VI Preface
The second part of the present notes covers the content of the subsequent
school on ‘Concentration Phenomena for Variational Problems’ held at the
Department of Mathematics of the University of Rome ‘La Sapienza’, Sep-
tember1–5,2003(co-organizedbyA.Braides,whichexplainswhyheappears
both as an editor and as a contributor). Scope of the School was to present
different problems in the Calculus of Variations depending on a small para-
meterε,thatexhibitadramatic‘changeoftype’asthisparametertendsto0,
that is best described by the ‘concentration’ of some quantity at some lower-
dimensionalset.ThecoursesofSylviaSerfatyandDidierSmetstreatthecase
of Ginzburg-Landau energies. In a two-dimensional setting it is known that
the concentration of Jacobians of minimizers at points can be interpreted as
the arising of ‘vortices’. A novel method envisaged by Sandier and Serfaty
shows how the limit motion of these vortices can be described by making use
of Γ-convergence. On the other hand, Smets’s course focuses on the informa-
tion that can be obtained by looking at the fine behaviour of solutions of the
Allen-Cahn equations, and concerns the motion in any dimension. The use of
Γ-convergence as a way to describe the concentration of maximizers of prob-
lemswithsub-criticalgrowthisalsothesubjectofthethirdcoursebyAdriana
Garroni. Here the concentrating quantity is not a Jacobian (the problem is
scalar), but a suitable scaling of the square of the gradient of the maximizer,
that converges as a measure to a sum of Dirac masses. This phenomenon has
been previously described by means of the Concentration-Compactness alter-
native, and this ‘version’ by Γ-convergence gives a new interpretation of the
results.
ThecourseofSylviaSerfaty,originallyprogrammedforthisschool,hadto
be postponed to a subsequent spin-off ‘School on Geometric Evolution Prob-
lems’ held at the Department of Mathematics of the University of Rome ‘Tor
Vergata’, January 26–28, 2004 (with the same organizing team, and an ad-
ditional course by Giovanni Bellettini) but is considered essentially part of
the September 2003 School, and that is why it is included here. Other two
courses, whose notes are not presented here, were held at the School by Gio-
vanniAlbertiandHalilMeteSoner.ThecourseofSonerfollowedthenotesof
a previous school and can be found in his Lecture Notes ‘Variational and dy-
namic problems for the Ginzburg-Landau functional. Mathematical aspects
of evolving interfaces’ (Lecture Notes in Math. 1812, Springer, 2003, 177–
233). Alberti’s presentation is partly covered by his review paper ‘A varia-
tional convergence result for Ginzburg-Landau functionals in any dimension’
(Boll. Un. Mat Ital. 4 (2001), 289–310). As a final acknowledgement, it must
be mentioned that these schools had been additionally jointly sponsored by
the Rome and Milan Units of the National Project ‘Calculus of Variations’.
Rome and Turin, Andrea Braides
February 2006 Valeria Chiado` Piat
Contents
Part I Problems with multiple scales
From discrete systems to continuous variational problems: an
introduction
Andrea Braides and Maria Stella Gelli.............................. 3
1 Discrete problems with limit energies defined on Sobolev spaces .... 6
1.1 Discrete functionals ...................................... 6
1.2 Equivalent energies on Sobolev functions.................... 7
1.3 Convex energies ......................................... 8
1.4 Non-convex energies with superlinear growth ................ 11
1.5 A general convergence theorem ............................ 19
1.6 Convergence of minimum problems......................... 21
1.7 More examples .......................................... 28
1.8 Energies depending on second-difference quotients ........... 35
2 Limit energies on discontinuous functions: two examples........... 36
2.1 The Blake-Zisserman model ............................... 36
2.2 Second-difference interactions ............................. 51
2.3 Lennard-Jones potentials ................................. 52
2.4 Renormalization-group approach........................... 58
3 General convergence results.................................... 58
3.1 Functions of bounded variation ............................ 58
3.2 Nearest-neighbour interactions ............................ 60
3.3 Long-range interactions................................... 73
References ...................................................... 75
Relaxation for bulk and interfacial energies
Anneliese Defranceschi ........................................... 79
1 Introduction ................................................. 79
2 Lower semicontinuity - The direct method of the Calculus of
Variations ................................................... 81
3 Relaxation .................................................. 83
VIII Contents
4 Preliminaries.Functionsofboundedvariationandspecialfunctions
of bounded variation.......................................... 85
4.1 Notation................................................ 85
4.2 Measure spaces.......................................... 86
4.3 Functions of bounded variation ............................ 88
4.4 Sets of finite perimeter ................................... 90
4.5 Structure of BV functions. Approximate discontinuity points
and approximate jump points ............................. 91
4.6 Special functions of bounded variation...................... 92
4.7 BV (SBV) functions in one dimension ..................... 92
5 Lower semicontinuity and relaxation in one dimension............. 93
5.1 A characterization of l.s.c. for functionals defined on SBV(I).
Relaxation on BV(I)..................................... 93
5.2 A sufficient condition for l.s.c. for functionals defined on
SBV(I). Relaxation in BV(I).............................100
6 Lower semicontinuity and relaxation in higher dimension ..........106
7 Lower semicontinuity and relaxation in higher dimension and for
vector-valued functions........................................113
7.1 Upper bound for the relaxed functional.....................119
7.2 De Giorgi-Letta criterion to prove measure properties of the
relaxed functional........................................121
7.3 Integral representation ...................................125
7.4 Proof of the main theorem ................................126
8 Relaxation in higher dimension (isotropy assumptions) ............129
9 Appendix ...................................................132
9.1 Appendix A: Lower semicontinuous envelope ................132
9.2 Appendix B: Proof of (73) ................................135
9.3 Appendix C: Proof of Step 3 in Proposition 67 ..............135
References ......................................................136
Convergence of Dirichlet forms on fractals
Roberto Peirone .................................................139
1 Introduction .................................................139
2 Construction of an energy on the Gasket ........................141
3 Dirichlet forms on finitely ramified fractals ......................152
4 Main properties of renormalization and harmonic extension........164
5 Homogenization on the Gasket .................................173
6 Homogenization on general fractals .............................179
References ......................................................188
Homogenization in perforated domains
Gregory A. Chechkin .............................................189
1 Appearence of a “term etrange”. Dirichlet problems in domains
with low concentration of holes.................................190
1.1 Basic notation and setting of the problem...................190
Contents IX
1.2 The homogenization theorem..............................191
2 Homogenization problems in perforated domain with oscillating
Fourier boundary conditions ...................................193
2.1 Basic notation...........................................193
2.2 Formal asymptotic analysis of the problem..................194
2.3 Main estimates and results. ...............................197
2.4 Auxiliary results.........................................199
2.5 Justification of asymptotics ...............................203
References ......................................................207
Homogenization of random non stationary parabolic operators
Andrey Piatnitski ................................................209
1 The setup ...................................................210
2 Factorization of the equation...................................213
3 A priori estimates for the factorized equation ....................214
4 Auxiliary problems ...........................................215
5 Homogenization of the factorized equation.......................218
6 Homogenization of the original equation.........................223
7 Equations with diffusion driving process.........................225
References ......................................................230
Part II Problems with concentration
Γ-convergence for concentration problems
Adriana Garroni.................................................233
1 Introduction .................................................233
2 Two classical examples........................................234
2.1 Sobolev inequality .......................................235
2.2 Capacity and isoperimetric inequality for the capacity ........238
3 The general problem..........................................242
3.1 Variational formulation...................................242
3.2 Generalized Sobolev inequality ............................244
3.3 Concentration ...........................................244
3.4 Γ+-convergence .........................................245
3.5 The concentration result in terms of Γ+-convergence .........246
4 Identification of the concentration point .........................252
4.1 The Green’s function and the Robin function................253
4.2 Asymptotic expansion in Γ+-convergence ...................256
4.3 The result ..............................................257
4.4 Harmonic transplantation.................................260
5 Irregular domains ............................................263
References ......................................................265
X Contents
Gamma-convergence of gradient flows and applications to
Ginzburg-Landau vortex dynamics
Sylvia Serfaty ...................................................267
1 Introduction .................................................267
1.1 Presentation of the Ginzburg-Landau model.................267
1.2 Gamma-convergence .....................................269
1.3 Γ-convergence of Ginzburg-Landau ........................271
2 The abstract result for Γ-convergence of gradient-flows............275
2.1 The abstract situation....................................275
2.2 The result ..............................................276
2.3 Interpretation ...........................................279
2.4 Idea of the proof.........................................279
2.5 Application to Ginzburg-Landau...........................281
2.6 Remarks................................................282
3 Proof of the additional conditions for Ginzburg-Landau ...........284
3.1 A product-estimate for Ginzburg-Landau ...................284
3.2 Idea of the proof.........................................285
3.3 Application to the dynamics ..............................286
3.4 Proof of the construction 2(cid:1)) ..............................287
4 Extensions of the method .....................................289
4.1 Collisions ...............................................289
4.2 Second order questions - stability issues ....................290
References ......................................................291
PDE analysis of concentrating energies for the Ginzburg-
Landau equation
Didier Smets ....................................................293
1 Introduction .................................................293
2 The monotonicity formula .....................................294
2.1 Scaling properties........................................294
2.2 The monotonicity formula ................................295
2.3 Consequences for µ∗ .....................................297
3 Regularity theorems for (GL) .................................299
ε
3.1 Bochner, small energy regularity and clearing-out ............299
3.2 Consequences for µ∗ .....................................306
4 Potential and modulus estimates ...............................308
5 Σ is a minimal surface .......................................311
µ
5.1 Classical and weak notions of mean curvature ...............311
5.2 Deriving the curvature equation ...........................312
References ......................................................314
Index..........................................................315