The monge ampère equation second edition

Progress in Nonlinear Differential Equations
and Their Applications
89

Cristian E. Gutiérrez

The MongeAmpère
Equation
Second Edition



Progress in Nonlinear Differential
Equations and Their Applications
Volume 89
Editor
Haim Brezis
Université Pierre et Marie Curie, Paris, France
Technion – Israel Institute of Technology, Haifa, Israel
Rutgers University, New Brunswick, NJ, USA
Editorial Board
Antonio Ambrosetti, Scuola Internationale Superiore di Studi
Avanzati, Trieste, Italy
A. Bahri, Rutgers University, New Brunswick, NJ, USA
Felix Browder, Rutgers University, New Brunswick, NJ, USA
Luis Caffarelli, The University of Texas, Austin, TX, USA
Jean-Michel Coron, University Pierre et Marie Curie, Paris, France
Lawrence C. Evans, University of California, Berkeley, CA, USA
Mariano Giaquinta, University of Pisa, Italy
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh, PA,
USA

Sergiu Klainerman, Princeton University, NJ, USA
Robert Kohn, New York University, NY, USA
P. L. Lions, Collège de France, Paris, France
Jean Mawhin, Université Catholique de Louvain, Louvain-la-Neuve, Belgium
Louis Nirenberg, New York University, NY, USA
Paul Rabinowitz, University of Wisconsin, Madison, WI, USA
John Toland, Isaac Newton Institute, Cambridge, UK

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Cristian E. Gutiérrez

The Monge-Ampère
Equation
Second Edition


Cristian E. Gutiérrez
Department of Mathematics
Temple University
Philadelphia, Pennsylvania, USA

ISSN 1421-1750
ISSN 2374-0280 (electronic)
Progress in Nonlinear Differential Equations and Their Applications
ISBN 978-3-319-43372-1
ISBN 978-3-319-43374-5 (eBook)
DOI 10.1007/978-3-319-43374-5
Library of Congress Control Number: 2016950029
Mathematics Subject Classification (2010): 35J60, 35J65, 53A15, 52A20

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Preface to the Second Edition

A considerable amount of material has been added to this edition. It contains two
new chapters: Chapter 7 on the linearized Monge–Ampère equation and Chapter 8
on Hölder estimates for second derivatives of solutions to the Monge–Ampère
equation. In addition, a set of 31 exercises is added to Chapter 1. The notes at the end
of each chapter have been updated to reflect new developments since the publication
of the first edition in 2001. Several misprints and errors from the first edition have
been corrected, and more clarifications have been added.
Chapter 8 is written in collaboration with Qingbo Huang and Truyen Nguyen
to whom I am also extremely grateful for numerous suggestions that improved the

presentation.
I thank Farhan Abedin for carefully reading Chapters 1, 5, and 7 and for
providing several suggestions that made some proofs more clear.
I hope this new edition will continue serving to stimulate research on the Monge–
Ampère equation, its connections with several areas, and its applications.
Moorestown, NJ, USA
April 2016

Cristian E. Gutiérrez

v


Preface to the First Edition

In recent years, the study of the Monge–Ampère equation has received considerable
attention, and there have been many important advances. As a consequence, there
is nowadays much interest in this equation and its applications. This volume tries
to reflect these advances in an essentially self-contained systematic exposition of
the theory of weak solutions, including recent regularity results by L. A. Caffarelli.
The theory has a geometric flavor and uses some techniques from harmonic analysis
such us covering lemmas and set decompositions. An overview of the contents of
the book is as follows:
We shall be concerned with the Monge–Ampère equation, which for a smooth
function u, is given by
det D2 u D f :

(0.0.1)

There is a notion of generalized or weak solution to (0.0.1): for u convex in a

domain , one can define a measure Mu in
such that if u is smooth, then Mu
has density det D2 u: Therefore, u is a generalized solution of (0.0.1) if Mu D f :
The notion of generalized solution is based on the notion of normal mapping, and in
Chapter 1 we begin with these two concepts, introduced by A. D. Aleksandrov, and
we describe their basic properties. The notion of viscosity solution is also considered
and compared with that of generalized solution. We also introduce several maximum
principles that are fundamental in the study of the Monge–Ampère operator. The
Dirichlet problem for Monge–Ampère is then solved in the class of generalized
solutions in Sections 1.5 and 1.6. Chapter 1 concludes with the concept of ellipsoid
of minimum volume which is of particular importance in developing the theory of
cross sections in Chapter 3.
In Chapter 2, we present the Harnack inequality of Krylov–Safonov for nondivergence elliptic operators in view of some ideas used to study the linearized
Monge–Ampère equation. This illustrates these ideas in a case that is simpler than
that of the linearized Monge–Ampère operator.

vii


viii

Preface to the First Edition

Chapter 3 presents the theory of cross sections of weak solutions to the Monge–
Ampère equation, and we prove several geometric properties that are needed in
the subsequent chapters. The cross sections of u are the level sets of the convex
function u minus a supporting hyperplane. Of special importance is the doubling
condition (3.1.1) for the measure Mu that permits us, from the characterization
given in Theorem 3.3.5, to determine invariance properties for the shapes of
cross sections that are valid under appropriate normalizations using ellipsoids of

minimum volume. A typical situation is when the measure Mu satisfies
jEj Ä Mu.E/ Ä ƒ jEj;

(0.0.2)

for some positive constants ; ƒ and for all Borel subsets E of the convex domain
: The inequalities (0.0.2) resemble the uniform ellipticity condition for linear
operators. The results proved in this chapter permit us to work with the cross
sections as if they were Euclidean balls and to establish the covering lemmas needed
later for the regularity theory in Chapters 4–6.
Chapter 4 concerns an application of the properties of the sections: a result
of Jörgens–Calabi–Pogorelov–Cheng and Yau about the characterization of global
solutions of Mu D 1.
Chapter 5 contains Caffarelli’s C1;˛ estimates for weak solutions. A fundamental
geometric result is Theorem 5.2.1 about the extremal points of the set where a
solution u equals a supporting hyperplane.
Finally, in Chapter 6, we present the W 2;p estimates for the Monge–Ampère equation recently developed by Caffarelli and extend classical estimates of Pogorelov.
The main result here is Theorem 6.4.2.
We have included bibliographical notes at the end of each chapter.

Acknowledgments
It is a pleasure to thank all the people who assisted me during the preparation
of this book. I am particularly indebted to L. A. Caffarelli for inspiration, many
discussions, and for his collaboration. I am very grateful to Qingbo Huang for
innumerable enlightening discussions on most topics in this book, for many
suggestions, and corrections, and for his collaboration. I am also very grateful to
several friends and students for carefully reading various chapters of the manuscript:
Shif Berhanu, Giuseppe Di Fazio, David Hartenstine, and Federico Tournier. They
have made many helpful comments, suggestions and corrections that improved the
presentation. I would especially like to thank L. C. Evans for his encouragement and

suggestions.
This book encompasses the contents of a graduate course at Temple University,
and some chapters have been used in short courses at the Università di Bologna,
Universidad de Buenos Aires, and Universidad Autónoma de Madrid. I would like to
thank these institutions and all my friends there for the kind hospitality and support.


Preface to the First Edition

ix

The research connected with the results in this volume was supported in part by
the National Science Foundation, and I wish to thank this institution for its support.
Cherry Hill, NJ, USA
September 2000

Cristian E. Gutiérrez


Notation

Du denotes the gradient of the function u:

Â

D2 u.x/ denotes the Hessian of the function u, i.e., D2 u.x/ D
1 Ä i; j Ä n.
Rn , u W
t x C .1 t/ y 2

! R is convex if for all 0 Ä t Ä 1 and any x; y 2
we have
u.t x C .1

t/ y/ Ä tu.x/ C .1

Ã
@2 u.x/
;
@xi @xj
such that

t/u.y/:

Given a set E, E .x/ denotes the characteristic function of E.
jEj denotes the Lebesgue measure of the set E.
BR .x/ denotes the Euclidean ball centered at x with radius R.
!n denotes the measure of the unit ball in Rn .
C. / denotes the class of real-valued functions that are continuous in :
Given a positive integer k, Ck . / denotes the class of real-valued functions that
are continuously differentiable in up to order k.
If Ek is a sequence of sets, then
1
E D lim sup En D \1
nD1 [kDn Ek I

1
E D lim inf En D [1
nD1 \kDn Ek I
n!1

n!1
E

.x/ D lim sup
n!1

En .x/I

E

.x/ D lim inf
n!1

En .x/:

The real-valued function u is harmonic in the open set
Rn if u 2 C2 . / and
Pn @2 u.x/
u.x/ D iD1
D 0 in :
@xi2
If
Rn is a bounded and measurable set, the center of mass or barycenter of
is the point x defined by
x D

1
j j

Z
x dx:

xi


xii

Notation

If A B Rn and AN B, then we write A b B:
If a; b 2 R, then a _ b D maxfa; bg:
If E is a set, then P.E/ denotes the class of all subsets of E.
If Q Rn is a cube and ˛ > 0, then ˛ Q denotes the cube concentric with Q but
with edge length equals ˛ times the edge length of Q.


Contents

1

Generalized Solutions to Monge–Ampère Equations . . . . . . . . . . . . . . . . . . . .
1.1
The Normal Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Properties of the Normal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4

Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Aleksandrov’s Maximum Principle. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Aleksandrov–Bakelman–Pucci’s Maximum Principle . . . . . .
1.4.3 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6
The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7
Return to Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8
Ellipsoids of Minimum Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
6
8
10
11
13
17
18
21
26
28
31

39

2

Uniformly Elliptic Equations in Nondivergence Form. . . . . . . . . . . . . . . . . . .
2.1
Critical Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Estimate of the Distribution Function of Solutions . . . . . . . . . . . . . . . . . .
2.3
Harnack’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
48
51
54

3

The Cross-Sections of Monge–Ampère . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Properties of the Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The Monge–Ampère Measures Satisfying (3.1.1) . . . . . . . . . . .

3.3.2 The Engulfing Property of the Sections . . . . . . . . . . . . . . . . . . . . . .
3.3.3 The Size of Normalized Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
55
57
63
63
68
70
75

xiii


xiv

Contents

4

Convex Solutions of det D2 u D 1 in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Pogorelov’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Interior Hölder Estimates of D2 u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
C˛ Estimates of D2 u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77
77
81
84
89

5

Regularity Theory for the Monge–Ampère Equation. . . . . . . . . . . . . . . . . . . .
5.1
Extremal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
A result on extremal points of zeroes of solutions
to Monge–Ampère . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
A Strict Convexity Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
C1;˛ Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 C1;˛ Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 A Generalization of Formula (5.5.1) . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
91

93
96
101
106
111
112
121

6

W 2 ;p
6.1
6.2
6.3
6.4
6.5
6.6
6.7

Estimates for the Monge–Ampère Equation . . . . . . . . . . . . . . . . . . . . . . . .
Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tangent Paraboloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density Estimates and Power Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lp Estimates of Second Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proof of the Covering Theorem 6.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Regularity of the Convex Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123
123

127
129
137
141
148
150

7

The Linearized Monge–Ampère Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Normalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
Critical Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Double Section Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 A1 Condition on Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Behavior of nonnegative Solutions in Expanded Sections . .
7.5
A Calderón-Zygmund Type Decomposition for Sections . . . . . . . . . . .
7.6
Power Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7
Interior Harnack’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

153
155
156
161
163
165
172
181
187
191

8

Interior Hölder Estimates for Second Derivatives . . . . . . . . . . . . . . . . . . . . . . .
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Interior C2;˛ Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193
193
193
209

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Chapter 1

Generalized Solutions to Monge–Ampère
Equations

1.1 The Normal Mapping
Let
be an open subset of Rn and u W
! R. Given x0 2 , a supporting
hyperplane to the function u at the point .x0 ; u.x0 // is an affine function `.x/ D
u.x0 / C p .x x0 / such that u.x/ `.x/ for all x 2 :
Definition 1.1.1. The normal mapping of u, or subdifferential of u, is the set-valued
function @u W ! P.Rn / defined by
@u.x0 / D fp W u.x/
Given E

, we define @u.E/ D

u.x0 / C p .x
S
x2E

x0 /;

for all x 2

g:

@u.x/:

The set @u.x0 / may be empty. Let S D fx 2
W @u.x/ ¤ ;g: If u 2 C1 . /
and x 2 S, then @u.x/ D Du.x/, the gradient of u at x, which means that when u
is differentiable the normal mapping is the gradient; see Exercise 11. If u 2 C2 . /
and x 2 S, then the Hessian of u is nonnegative definite, that is D2 u.x/ 0: This
means that if u is C2 , then S is the set where the graph of u is concave up. Indeed,
1
by Taylor’s Theorem u.x C h/ D u.x/ C Du.x/ h C hD2 u. /h; hi, where lies
2
on the segment between x and x C h. Since u.x C h/
u.x/ C Du.x/ h for all h
sufficiently small, the claim follows.
Example 1.1.2. It is useful to calculate the normal mapping of the function u whose
jx x0 j
graph is a cone in RnC1 : Let D BR .x0 / in Rn , h > 0 and u.x/ D h
: The
R
nC1
with vertex at the point
graph of u, for x 2 , is an upside-down right-cone in R
.x0 ; 0/ and base on the hyperplane xnC1 D h. We shall show that

© Springer International Publishing 2016
C.E. Gutiérrez, The Monge-Ampère Equation, Progress in Nonlinear Differential
Equations and Their Applications 89, DOI 10.1007/978-3-319-43374-5_1

1


2

1 Generalized Solutions to Monge–Ampère Equations

@u.x/ D

8
h x
ˆ
ˆ
ˆ
< R jx

x0
;
x0 j

ˆ
ˆ
ˆ
:B .0/;
h=R

for 0 < jx

x0 j < R,

for x D x0 :

If 0 < jx

x0 j < R, then the value of @u follows by calculating the gradient. By the
h
definition of normal mapping, p 2 @u.x0 / if and only if jx x0 j p .x x0 / for
R
h
p
all x 2 BR .x0 /. If p ¤ 0 and we pick x D x0 C R , then jpj Ä : It is clear that
jpj
R
h
jpj Ä implies p 2 @u.x0 /.
R

1.1.1 Properties of the Normal Mapping
Lemma 1.1.3. If
compact.

Rn is open, u 2 C. / and K

is compact, then @u.K/ is

Proof. Let fpk g @u.K/ be a sequence. We claim that pk are bounded. For each k
there exists xk 2 K such that pk 2 @u.xk /; that is u.x/ u.xk / C pk .x xk / for all
x 2 : Since K is compact, Kı D fx W dist.x; K/ Ä ıg is compact and contained in
for all ı sufficiently small, and we may assume by passing if necessary through a
subsequence that xk ! x0 . Then xk C ıw 2 Kı , and u.xk C ıw/ u.xk / C ıpk w
pk
, then we get maxKı u.x/
for all jwj D 1 and for all k. If pk ¤ 0 and w D
jpk j

minK u.x/ C ıjpk j; for all k. Since u is locally bounded, the claim is proved. Hence
there exists a convergent subsequence pkm ! p0 . We claim that p0 2 @u.K/: We
shall prove that p0 2 @u.x0 /: We have u.x/ u.xkm / C pkm .x xkm / for all x 2
and, since u is continuous, by letting m ! 1 we obtain u.x/ u.x0 / C p0 .x x0 /
for all x 2 . This completes the proof of the lemma.
Remark 1.1.4. We note that the proof above shows that if u is only locally bounded
in , then @u.E/ is bounded whenever E is bounded with E
.
Remark 1.1.5. We note that given x0 2 , the set @u.x0 / is convex. However, if K
is convex and K
, then the set @u.K/ is not necessarily convex. An example is
2
given by u.x/ D ejxj and K D fx 2 Rn W jxi j Ä 1; i D 1; : : : ; ng: The set @u.K/ is
a star-shaped symmetric set around the origin that is not convex, see Figure 1.1.
Lemma 1.1.6. If u is a convex function in
and K
is compact, then u is
uniformly Lipschitz in K, that is, there exists a constant C D C.u; K/ such that
ju.x/ u.y/j Ä Cjx yj for all x; y 2 K:
Proof. Since u is convex, u has a supporting hyperplane at any x 2
supfjpj W p 2 @u.K/g. By Lemma 1.1.3, C < 1: If x 2 K, then u.y/

: Let C D
u.x/ C p


1.1 The Normal Mapping

3

Fig. 1.1 @u.K/

.y x/ for p 2 @u.x/ and for all y 2 : In particular, if y 2 K, then u.y/
jpjjy xj: By reversing the roles of x and y we get the lemma.
Lemma 1.1.7. If
tiable a.e. in :

is open and u is Lipschitz continuous in

u.x/

, then u is differen-

Proof. See [EG92, p. 81].
Lemma 1.1.8. If u is convex or concave in

, then u is differentiable a.e. in

:

Proof. Follows immediately from Lemmas 1.1.6 and 1.1.7.
Remark 1.1.9. A deep result of Busemann–Feller–Aleksandrov establishes that
any convex function in has second order derivatives a.e., see [EG92, p. 242] and
[Sch93, pp. 31–32].
Definition 1.1.10. The Legendre transform of the function u W
function u W Rn ! R defined by
u .p/ D sup .x p

! R is the

u.x// :

x2

Remark 1.1.11. If
is convex in Rn .

is bounded and u is bounded in

, then u is finite. Also, u

Lemma 1.1.12. If is open and u is a continuous function in , then the set of
points in Rn that belong to the image by the normal mapping of more than one point
of has Lebesgue measure zero. That is, the set
S D fp 2 Rn W there exist x; y 2

, x ¤ y and p 2 @u.x/ \ @u.y/g

has measure zero. This also means that the set of supporting hyperplanes that touch
the graph of u at more than one point has measure zero.


4

1 Generalized Solutions to Monge–Ampère Equations

Proof. We may assume that is bounded because otherwise we write D [k k ,
where k
,
kC1 are open and

k are compact. If p 2 S, then there exist x; y 2
x ¤ y and u.z/ u.x/ C p .z x/; u.z/ u.y/ C p .z y/ for all z 2 . Since k
increases, x; y 2 m for some m and obviously the previous inequalities hold true
for z 2 m . That is, if
Sm D fp 2 Rn W there exist x; y 2

, x ¤ y and p 2 @.uj

m /.x/

\ @.uj

m /.y/g

we have p 2 Sm , i.e., S [m Sm and we then show that each Sm has measure zero.
Let u be the Legendre transform of u. By Remark 1.1.11 and Lemma 1.1.8, u
is differentiable a.e. Let E D fp W u is not differentiable at pg: We shall show that
fp 2 Rn W there exist x; y 2

, x ¤ y and p 2 @u.x/ \ @u.y/g

E:

In fact, if p 2 @u.x1 / \ @u.x2 / and x1 ¤ x2 , then u .p/ D xi p u.xi /, i D 1; 2: Also
u .z/ xi z u.xi / and so u .z/ u .p/ C xi .z p/ for all z, i D 1; 2. Hence if
u were differentiable at p, we would have Du .p/ D xi ; i D 1; 2: This completes
the proof of the lemma.
Theorem 1.1.13. If

is open and u 2 C. /, then the class

S D fE

W @u.E/ is Lebesgue measurableg

is a Borel -algebra. The set function Mu W S ! R defined by
Mu.E/ D /j

(1.1.1)

is a measure, finite on compacts, that is called the Monge–Ampère measure
associated with the function u.
Proof. By Lemma 1.1.3, the class S contains all compact subsets of . Also, if Em
is any sequence of subsets of , then @u .[m Em / D [m @u.Em /: Hence, if Em 2 S,
m D 1; 2; : : : ; then [m Em 2 S: In particular, we may write
D [m Km with Km
compacts and we obtain 2 S. To show that S is a -algebra it remains to show
that if E 2 S, then n E 2 S: We use the following formula, which is valid for any
set E
:
@u.

n E/ D .@u. / n @u.E// [ .@u.

n E/ \ @u.E// :

(1.1.2)

By Lemma 1.1.12, j@u. n E/ \ @u.E/j D 0 for any set E. Then from (1.1.2) we get
n E 2 S when E 2 S.

We now show that Mu is -additive. Let fEi g1
iD1 be a sequence of disjoint sets in
S and set @u.Ei / D Hi . We must show that
1
ˇ
ˇ X
ˇ@u [1 Ei ˇ D
jHi j:
iD1
iD1


1.1 The Normal Mapping

5

1
Since @u [1
iD1 Ei D [iD1 Hi , we shall show that
1
ˇ 1 ˇ X
ˇ[ Hi ˇ D
jHi j:
iD1

(1.1.3)

iD1

We have Ei \ Ej D ; for i ¤ j. Then by Lemma 1.1.12 jHi \ Hj j D 0 for i ¤ j: Let

us write
[1
iD1 Hi D H1 [ .H2 n H1 / [ .H3 n .H2 [ H1 // [ .H4 n .H3 [ H2 [ H1 // [

;

where the sets on the right-hand side are disjoint. Now
Hn D ŒHn \ .Hn

1

[ Hn

2

[

Then by Lemma 1.1.12, jHn \ .Hn

[ H1 / [ ŒHn n .Hn
1

[ Hn

jHn j D jHn n .Hn

1

2

[

[ Hn

1

[ Hn

2

[

[ H1 / :

[ H1 /j D 0 and we obtain
2

[

[ H1 /j:

Consequently (1.1.3) follows, and the proof of the theorem is complete.
Example 1.1.14. If u 2 C2 . / is a convex function, then the Monge–Ampère
measure Mu associated with u satisfies
Z
Mu.E/ D det D2 u.x/ dx;
(1.1.4)
E

for all Borel sets E

: To prove (1.1.4), we use the following result:

Theorem 1.1.15 (Sard’s Theorem, See [Mil97]). Let
Rn be an open set and
n
1
0
g W ! R a C function in . If S0 D fx 2 W det g .x/ D 0g, then jg.S0 /j D 0:
We first notice that since u is convex and C2 . /, then Du is one-to-one on the
set A D fx 2
W D2 u.x/ > 0g: Indeed, let x1 ; x2 2 A with Du.x1 / D Du.x2 /.
By convexity u.z/
u.xi / C Du.xi / .z xi / for all z 2 , i D 1; 2: Hence
u.x1 / u.x2 / D Du.x1 / .x1 x2 / D Du.x2 / .x1 x2 /: By the Taylor formula we
can write
u.x1 / D u.x2 / C Du.x2 / .x1 x2 /
Z 1
C
thD2 u .x2 C t.x1 x2 // .x1
0

x2 /; x1

x2 i dt:

Therefore the integral is zero and the integrand must vanish for 0 Ä t Ä 1. Since
x2 2 A, it follows that x2 C t.x1 x2 / 2 A for t small. Therefore x1 D x2 :
If u 2 C2 . /, then g D Du 2 C1 . /: We have Mu.E/ D jDu.E/j and
Du.E/ D Du.E \ S0 / [ Du.E n S0 /:

6

1 Generalized Solutions to Monge–Ampère Equations

Since E
Rn is a Borel set, E \ S0 and E n S0 are also Borel sets. Hence, by the
formula of change of variables and Sard’s Theorem,
Z
Mu.E/ D Mu.E \ S0 / C Mu.E n S0 / D

det D2 u.x/ dx D

EnS0

Z

det D2 u.x/ dx;

E

which shows (1.1.4).
Example 1.1.16. If u.x/ is the cone of Example 1.1.2, then the Monge–Ampère
measure associated with u is Mu D jBh=R j ıx0 ; where ıx0 denotes the Dirac delta
at x0 :

1.2 Generalized Solutions
Definition 1.2.1. Let be a Borel measure defined in , an open and convex subset
of Rn . The convex function u 2 C. / is a generalized solution, or Aleksandrov

solution, to the Monge–Ampère equation
det D2 u D
if the Monge–Ampère measure Mu associated with u defined by (1.1.1) equals :
See Exercise 31.
The following lemma implies that the notion of generalized solution is closed
under uniform limits. That is, if uk are generalized solutions to det D2 u D in
and uk ! u uniformly on compact subsets of , then u is also a generalized solution
to det D2 u D in :
Lemma 1.2.2. Let uk 2 C. / be convex functions such that uk ! u uniformly on
compact subsets of :
We have
(i) If K

is compact, then
lim sup @uk .K/ Â @u.K/;
k!1

and by Fatou’s lemma
lim sup j@uk .K/j Ä /j:
k!1

(ii) If U is open such that U

, then
@u.U/ Â lim inf @uk .U/;
k!1


1.2 Generalized Solutions

7

where the inequality holds for almost every point of the set on the left-hand
side1 , and by Fatou’s lemma
/j Ä lim inf j@uk .U/j:
k!1

Proof. (i) If p 2 lim supk!1 @uk .K/, then for each n there exist kn and xkn 2 K
such that p 2 @ukn .xkn /: By selecting a subsequence xj of xkn we may assume
that xj ! x0 2 K. On the other hand,
uj .x/

uj .xj / C p .x

xj /;

8x 2

;

and by letting j ! 1, by the uniform convergence of uj on compacts we get
u.x/

u.x0 / C p .x

x0 /;

8x 2

;

that is p 2 @u.x0 /.
(ii) Let S D fp W p 2 @u.x1 / \ @u.x2 / for some x1 ; x2 2 , x1 ¤ x2 g: By
Lemma 1.1.12, jSj D 0. Let U
be open and consider @u.U/ n S: If
p 2 @u.U/ n S, then there exists a unique x0 2 U such that p 2 @u.x0 / and
p … @u.x1 / for all x1 2 , x1 ¤ x0 : Let us first assume that U is compact, let
`.x/ D u.x0 / C p .x x0 /, and set ı D minfu.x/ `.x/ W x 2 @Ug > 0. From
the uniform convergence we have that ju.x/ uk .x/j < ı=2 for all x 2 U and
for all k k0 . Let
ık D maxf`.x/
x2U

uk .x/ C ı=2g:

Since x0 2 U, we have ık
`.x0 / uk .x0 / C ı=2 D u.x0 / uk .x0 / C ı=2 >
ı=2 C ı=2 D 0. We have ık D `.xk / uk .xk / C ı=2 for some xk 2 U. We
claim that xk … @U. Otherwise, by definition of ı, `.xk / u.xk / Ä ı and
so ık Ä ı=2, a contradiction. We claim that p is the slope of a supporting
hyperplane to uk at the point .xk ; uk .xk //. Indeed,
ık D u.x0 / C p .xk

x0 /

uk .xk / C ı=2

and so
uk .x/

uk .xk / C p .x

xk /

8x 2 U:

(1.2.1)

Since uk is convex in , U is open, and xk 2 U, it follows that (1.2.1)
k0 : This implies that
holds for all x 2 , that is p 2 @uk .xk / for all k
p 2 lim infk!1 @uk .U/:

The inclusion holds for @u.U/ n S, where S D fp W p 2 @u.x1 / \ @u.x2 / for some
x1 ; x2 2 , x1 ¤ x2 g:

1


8

1 Generalized Solutions to Monge–Ampère Equations

U

Finally, we remove the assumption
that U is compact. If U is open, and
S
, then we can write U D 1
U

jD1 j with Uj open and Uj compact. Then

@u.U/ D

1
[

@u.Uj / Â

jD1

1
[
jD1

1
[

lim inf @uk .Uj / Â
k!1

lim inf @uk .U/ D lim inf @uk .U/;

jD1

k!1

k!1

which completes the proof.

Lemma 1.2.3. If uk are convex functions in
such that uk ! u uniformly on
compact subsets of , then the associated Monge–Ampère measures Muk tend to
Mu weakly, that is
Z

Z
f .x/ dMuk .x/ !

f .x/ dMu.x/;

for every f continuous with compact support in

:

See Exercise 21 for a proof.

1.3 Viscosity Solutions
Definition 1.3.1. Let u 2 C. / be a convex function and f 2 C. /, f
0: The
function u is a viscosity subsolution (supersolution) of the equation det D2 u D f
in if whenever convex 2 C2 . / and x0 2
are such that .u
/.x/ Ä ( )
.u
/.x0 / for all x in a neighborhood of x0 , then we must have
det D2 .x0 /

.Ä/f .x0 /:

Remark 1.3.2. We claim that if u 2 C. / is convex,
local maximum at x0 2 , then
D2 .x0 /
In fact, since

2 C2 . / and u

has a

0:

2 C2 . /, we have

.x/ D .x0 / C D .x0 / .x

1
x0 / C hD2 .x0 /.x
2

Hence, for x close to x0 we get
u.x/ Ä .x/ C u.x0 /

.x0 /

D u.x0 / C D .x0 / .x

x0 /

x0 /; x

x0 i C o.jx

x0 j2 /:


1.3 Viscosity Solutions

9

C

1 2
hD .x0 /.x
2

x0 /; x

x0 i C o.jx

x0 j2 /:

Since u is convex, there exists p such that u.x/ u.x0 / C p .x x0 / for all x 2
Given jwj D 1 and > 0 small, by letting x x0 D w we obtain
p w Ä D .x0 / w C

1
2

2

:

hD2 .x0 /w; wi C o. 2 /:

Dividing this expression by , letting ! 0 and noting that the resulting inequality
holds for all jwj D 1 gives that p D D .x0 /: Hence hD2 .x0 /w; wi
0 and the
claim is proved.
Remark 1.3.3. We show that we may restrict the class of test functions used in the
definition of viscosity subsolution or supersolution to the class of strictly convex
quadratic polynomials. We shall first prove that, if the statement giving a strictly
convex quadratic polynomial and x0 2 such that .u
/.x/ Ä .u
/.x0 / for
all x in a neighborhood of x0 implies that
det D2 .x0 /

f .x0 /;

then u is a viscosity subsolution of the equation det D2 u D f in . To prove the
remark, let 2 C2 . / be convex such that u
has a local maximum at x0 2 .
We write
.x/ D .x0 / C D .x0 / .x
C

1 2
hD .x0 /.x
2

D P.x/ C o.jx

x0 /

x0 /; x

x0 i C o.jx

x0 j2 /

x0 j2 /:

Let > 0 and consider the quadratic polynomial P .x/ D P.x/ C jx
have

(1.3.1)
x0 j2 : We

D2 P .x0 / D D2 P.x0 / C 2 Id D D2 .x0 / C 2 Id;
and so the polynomial P is strictly convex. We have .x/ P .x/ D o.jx x0 j2 /
jx x0 j2 Ä 0 and so
P has a local maximum at x0 . Hence u P has a local
f .x0 /. By letting
maximum at x0 . Then det D2 P .x0 / D det D2 .x0 / C 2 Id
! 0, we obtain the desired inequality.
To prove the statement for supersolutions, let 2 C2 . / be convex such that
u
has a local minimum at x0 : If D2 .x0 / has some zero eigenvalue, then
2
det D .x0 / D 0 Ä f .x0 /: If all eigenvalues of D2 .x0 / are positive and P.x/ is

jx x0 j2 is strictly convex for all > 0
given by (1.3.1), then P .x/ D P.x/
sufficiently small. Proceeding as before, we now get that u P has a local minimum
at x0 and consequently det D2 .x0 / Ä f .x0 /:


10

1 Generalized Solutions to Monge–Ampère Equations

We now compare the two notions of solutions: generalized solutions and
viscosity solutions.
Proposition 1.3.4. If u is a generalized solution to Mu D f with f continuous, then
u is a viscosity solution.
Proof. Let 2 C2 . / be a strictly convex function such that u
has a local
maximum at x0 2 : We can assume that u.x0 / D .x0 /, then u.x/ < .x/ for all
0 < jx x0 j Ä ı. This can be achieved by adding rjx x0 j2 to and letting r ! 0
at the end.
Let m D minı=2Äjx x0 jÄı f .x/ u.x/g: We have m > 0. Let 0 < < m and
consider the set
S D fx 2 Bı .x0 / W u.x/ C > .x/g:
Bı=2 .x0 /.
If ı=2 Ä jx x0 j Ä ı, then .x/ u.x/ m and so x … S . Hence S
Let z 2 @S . Then there exist xn 2 S and xN n … S such that xn ! z and xN n ! z:
Hence u C D on @S . Since both functions are convex in S by Lemma 1.4.1,
we have that
@.u C /.S /

@ .S /:

Since u is a generalized solution, this implies that
Z

Z
f .x/ dx Ä j@.u C /.S /j Ä j@ .S /j D
S

det D2 .x/ dx:

S

By the continuity of f we obtain that det D2 .x0 / f .x0 /:
A similar argument shows that u is a viscosity supersolution.
We shall prove in Section 1.7 the converse of Proposition 1.3.4.

1.4 Maximum Principles
In this section we prove two maximum principles and a comparison principle for
the Monge–Ampère equation.
We begin with the following basic lemma.
Lemma 1.4.1. Let
Rn be a bounded open set, and u; v 2 C. /: If u D v on
@ and v u in , then
@v. /
see Figure 1.2.

@u. /I


1.4 Maximum Principles

Fig. 1.2 @v. /

11

@u. /

v

u

Proof. Let p 2 @v. /. There exists x0 2

such that

v.x0 / C p .x

v.x/

x0 /;

8x 2

:

Let
a D supfv.x0 / C p .x
x2

x0 /

u.x/g:

Since v.x0 / u.x0 /, it follows that a 0: We claim that v.x0 / C p .x x0 / a is
a supporting hyperplane to the function u at some point in : Since is bounded,
there exists x1 2 such that a D v.x0 / C p .x1 x0 / u.x1 / and so
u.x/

v.x0 / C p .x

x0 /

a D u.x1 / C p .x

x1 /

8x 2

:

We have
v.x1 /
Hence, if a > 0, then x1 … @
u.x/

v.x0 / C p .x1

x0 / D u.x1 / C a:

and so the claim holds in this case. If a D 0, then

v.x0 / C p .x

and consequently u.x0 / C p .x

x0 /

u.x0 / C p .x

x0 /

x0 / is a supporting hyperplane to u at x0 .

1.4.1 Aleksandrov’s Maximum Principle
The following estimate is fundamental in the study of the Monge–Ampère operator.


12

1 Generalized Solutions to Monge–Ampère Equations

Theorem 1.4.2 (Aleksandrov’s Maximum Principle). [Corrected] If
Rn is
a bounded, open, and convex set with diameter , and u 2 C. / is convex with
u D 0 on @ , then
ju.x0 /jn Ä Cn n 1 dist.x0 ; @ / j@u. /j;
for all x0 2

, where Cn is a constant depending only on the dimension n.

Proof. Fix x0 2 and let v be the convex function whose graph is the upside-down

cone with vertex .x0 ; u.x0 // and base , with v D 0 on @ . Since u is convex, v u
in . By Lemma 1.4.1
@v. /

@u. /:

To prove the theorem, we shall estimate the measure of @v. / from below. We first
notice that the set @v. / is convex. This is true because, if p 2 @v. /, then there
exists x1 2
such that p D @v.x1 /. If x1 ¤ x0 , since the graph of v is a cone,
then v.x1 / C p .x x1 / is a supporting hyperplane at x0 , that is p 2 @v.x0 /. So
@v. / D @v.x0 / and since @v.x0 / is convex we are done.
u.x0 /
:
Second, we notice that there exists p0 2 @v. / such that jp0 j D
dist.x0 ; @ /
This follows because is convex. Indeed, we take x1 2 @ such that jx1 x0 j D
dist.x0 ; @ / and let H be the unique supporting hyperplane to the set at x1 . The
uniqueness follows because Bjx1 x0 j .x0 /
; H has equation .x x1 / .x0
x1 / D 0. The hyperplane in RnC1 generated by H and the point .x0 ; u.x0 // is
a supporting hyperplane to v and has equation z D u.x0 / C p0 .x x0 / with
u.x0 /
.x0 x1 /.
p0 D
jx0 x1 j2
u.x0 /
is
Now notice that the ball B with center at the origin and radius


u.x0 /
contained in @v. /, and jp0 j
. Hence the convex hull of B and p0 is

contained in @v. / and it has measure greater than or equal to
Â
Cn
which proves the theorem.

u.x0 /


Ãn

1

jp0 j;