The Monge-Ampère equation and its applications

212 Pages·2017·1.778 MB·English

by  Alessio Figalli

Z U R I C H L E C T U R E S I N A D VA N C E D M AT H E M AT I C S Z U R I C H L E C T U R E S I N A D V A N C E D M A T H E M A T I C S
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Alessio Figalli T Alessio Figalli
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The Monge-Ampère Equation M
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The Monge–Ampère equation is one of the most important partial differential equations, p The Monge-Ampère
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appearing in many problems in analysis and geometry. r
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This monograph is a comprehensive introduction to the existence and regularity theory of q Equation and
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the Monge–Ampère equation and some selected applications; the main goal is to provide a
the reader with a wealth of results and techniques he or she can draw from to understand tio
current research related to this beautiful equation. n Its Applications
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The presentation is essentially self-contained, with an appendix wherein one can find d
precise statements of all the results used from different areas (linear algebra, convex it
geometry, measure theory, nonlinear analysis, and PDEs). s
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This book is intended for graduate students and researchers interested in nonlinear PDEs: p
explanatory figures, detailed proofs, and heuristic arguments make this book suitable for lic
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ISBN 978-3-03719-170-5
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Alessio Figalli
The Monge-Ampère
Equation and
Its Applications
Author:
Alessio Figalli
Department Mathematik
ETH Zürich
Ramistrasse 101
8092 Zürich
Switzerland
E-mail: [email protected]
2010 Mathematics Subject Classification: Primary: 35J96; secondary: 35B65, 35J60, 35J66, 35B45, 35B50,
35D05, 35D10, 35J65, 53A15, 53C45
Key words: Monge–Ampère equation, weak and strong solutions, existence, uniqueness, regularity

ISBN 978-3-03719-170-5
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9 8 7 6 5 4 3 2 1
ToMikaela

Preface
This book originates from a series of lectures given by the author at ETH Zürich
duringthefallof2014,intheframeworkofaNachdiplomvorlesung,ontheMonge–
Ampèreequationanditsapplications.
The Monge–Ampère equation is a fully nonlinear, degenerate elliptic equation
arising in several problems in the areas of analysis and geometry, such as the pre-
scribedGaussiancurvatureequation,affinegeometry,andoptimaltransportation. In
itsclassicalform,itconsistsofprescribingthedeterminantoftheHessianofaconvex
functionuinsidesomedomainΩ,thatis,
det(D2u)= f inΩ.
This is in contrast with the “model” elliptic equation ∆v = g, which prescribes the
traceoftheHessianofafunctionv. Thereareseveralboundaryconditionsthatone
mayconsiderforu,andinthisbookweshallfocusonDirichletboundaryconditions
prescribingthevalueofuon∂Ω.
Ourgoalistogiveacomprehensiveintroductiontotheexistenceandregularity
theory of the Monge–Ampère equation, and to show some selected applications.
Although some of the results contained here have already been discussed in the
classical book by Gutiérrez [61], recent developments in the theory have motivated
ustowriteanewbookonthesubject.
InthesamespiritasthelecturesgivenatETHZürich,thestructureofthisbook
followsa“historical”path. Moreprecisely,afterabriefintroductioninChapter1to
theMonge–Ampèreequationanditshistory,Chapter2isdedicatedtothetheoryof
weak solutions introduced by Alexandrov in the 1940s. This notion of solutions is
powerfulenoughtoallowonetoobtainexistenceanduniquenessofweaksolutions
with any nonnegative Borel measure as a right-hand side. Then in Chapter 3 we
address the issue of existence of global smooth solutions. This theory, developed
between the 1960s and 1980s, combines the continuity method and some interior a
priori estimates due to Pogorelov to show existence of smooth solutions when the
domain and the boundary data are smooth. The largest part of this book is devoted
to the interior regularity of weak solutions. Specifically, in Chapter 4 we study, in
detail,thegeometryofsolutions,mostlyinvestigatedbyCaffarelliinthe1990s,and
weproveinteriorC1,α,W2,p, andC2,α estimates. Finally, inChapter5wedescribe
someextensionsandgeneralizationsoftheresultsdescribedinthepreviouschapters.
Since this theory needs some general knowledge from different areas (linear
algebra, convex geometry, measure theory, nonlinear analysis, and PDEs), we have
decidedtowriteanappendixwherethereadercanfindprecisestatementsofallthe
viii Preface
results that we have used. Whenever possible we have included the proofs of such
results,andotherwisewehavegivenaprecisereference.
Bynomeansisthisbookintendedtocoverallthetopicsandrecentdevelopments
in the theory of the Monge–Ampère equation and its variants. Rather, our hope
and intent is that, after reading this book, the reader will be able to understand and
appreciatecontemporaryliteratureonthetopic.
The reader may notice that every chapter is divided into several sections and
subsections. Webelievethiswillfacilitatecomprehensionandhelpthereaderwhen
movingbetweendifferentresults. Also,forthesamereason,longproofsarealways
split into several steps. Finally, because of the geometric arguments presented, we
haveincludedseveralsupportingfigures. Wehopethatthereaderwillbenefitfrom
thispresentationstyle.
This book would not exist without the support and help of many friends and
colleagues. First,IhavebeenluckyenoughtohaveLuisCaffarelliasacolleagueand
departmentneighborformanyyears. HisbeautifulresultsonMonge–Ampèrehave
been a constant source of inspiration. Second, I have been fortunate to have Guido
DePhilippisasalong-timecollaborator;investigatingtheMonge–Ampèreequation
withhimhasbeentremendouslyinspiringandenjoyable.
IoweadebtofgratitudetoTristanRivière, MichaelStruwe, andtheentirestaff
atETH.Icannotbegintoexpressmyappreciationfortheirwarmhospitalityduring
my semester at ETH Zürich. Also, I wish to thank the faculty and students who
attendedmycourse; theirinterest, curiosity, andparticipationwasagreatsourceof
supportandmotivation. IamparticularlygratefultoCamilloDeLellisandThomas
Kappeler for their encouragement and to Maria Colombo and Mikaela Iacobelli for
theircarefullywrittennotes.
This book has benefited from valuable comments and suggestions by Connor
MooneyandNeilTrudinger. Furthermore,IhavetothankYashJhaveriforadetailed
andcarefulreadingofthewholemanuscript.
Finally,Iamverygratefulformyfamily’scontinuedencouragementandmywife
Mikaela’sconstantsupportandlove. Thankyouforalwaysbeingbymyside!
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 OnthedegeneracyoftheMonge–Ampèreequation . . . . . . . . . 1
1.2 Somehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Alexandrovsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 TheMonge–Ampèremeasure . . . . . . . . . . . . . . . . . . . . . 7
2.2 Alexandrovsolutions: Definitionandbasicproperties . . . . . . . . 11
2.3 TheDirichletproblem: Uniqueness . . . . . . . . . . . . . . . . . 17
2.4 TheDirichletproblem: Existence . . . . . . . . . . . . . . . . . . 20
2.5 C1 regularityin2-D . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Application1: TheMinkowskiproblemforcurvaturemeasures . . . 34
3 Smoothsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Existenceofsmoothsolutionsbythecontinuitymethod . . . . . . . 39
3.2 Pogorelov’scounterexampletointeriorregularity . . . . . . . . . . 53
3.3 Pogorelov’sinteriorestimatesandregularityofweaksolutions . . . 56
4 Interiorregularityofweaksolutions. . . . . . . . . . . . . . . . . . . . . 65
4.1 Sectionsandnormalizedsolutions . . . . . . . . . . . . . . . . . . 65
4.2 Onthestrictconvexityofsolutions . . . . . . . . . . . . . . . . . . 74
4.3 ALiouvilletheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Application2: Petty’stheorem . . . . . . . . . . . . . . . . . . . . 90
4.5 InteriorC1,α estimates . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Application3: Theoptimaltransportproblemwithquadraticcost . 95
4.7 Geometryofsections . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.8 InteriorW2,p estimates . . . . . . . . . . . . . . . . . . . . . . . . 111
4.9 Application4: Thesemigeostrophicequations . . . . . . . . . . . . 127
4.10 InteriorC2,α regularity . . . . . . . . . . . . . . . . . . . . . . . . 130
4.11 Wang’scounterexamples . . . . . . . . . . . . . . . . . . . . . . . 137
5 Furtherresultsandextensions. . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 FurtherresultsontheMonge–Ampèreequation . . . . . . . . . . . 141
5.2 ThelinearizedMonge–Ampèreequation . . . . . . . . . . . . . . . 148
5.3 AgeneralclassofMonge–Ampère-typeequations . . . . . . . . . . 150