The monge ampère equation second edition

It contains twonew chapters: Chapter7on the linearized Monge–Ampère equation and Chapter8on Hölder estimates for second derivatives of solutions to the Monge–Ampèreequation.. An overview

Progress in Nonlinear Differential Equations

and Their Applications

89

Cristian E Gutiérrez

The Ampère

Monge-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, BelgiumLouis Nirenberg, New York University, NY, USA

Paul Rabinowitz, University of Wisconsin, Madison, WI, USA

John Toland, Isaac Newton Institute, Cambridge, UK

More information about this series athttp://www.springer.com/series/4889

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

© Springer International Publishing 2001, 2016

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The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This book is published under the trade name Birkhäuser, www.birkhauser-science.com

The registered company is Springer International Publishing AG Switzerland

( www.birkhauser-science.com )

A considerable amount of material has been added to this edition It contains twonew chapters: Chapter7on the linearized Monge–Ampère equation and Chapter8

on Hölder estimates for second derivatives of solutions to the Monge–Ampèreequation In addition, a set of 31 exercises is added to Chapter1 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 havebeen corrected, and more clarifications have been added

Chapter8 is written in collaboration with Qingbo Huang and Truyen Nguyen

to whom I am also extremely grateful for numerous suggestions that improved thepresentation

I thank Farhan Abedin for carefully reading Chapters 1, 5, and 7 and forproviding 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

April 2016

v

Preface to the First Edition

In recent years, the study of the Monge–Ampère equation has received considerableattention, 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 ofthe theory of weak solutions, including recent regularity results by L A Caffarelli.The theory has a geometric flavor and uses some techniques from harmonic analysissuch us covering lemmas and set decompositions An overview of the contents ofthe book is as follows:

We shall be concerned with the Monge–Ampère equation, which for a smooth

function u, is given by

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 D2u : 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 inChapter1we begin with these two concepts, introduced by A D Aleksandrov, and

we describe their basic properties The notion of viscosity solution is also consideredand compared with that of generalized solution We also introduce several maximumprinciples that are fundamental in the study of the Monge–Ampère operator TheDirichlet problem for Monge–Ampère is then solved in the class of generalizedsolutions in Sections1.5and1.6 Chapter1concludes with the concept of ellipsoid

of minimum volume which is of particular importance in developing the theory ofcross sections in Chapter3

In Chapter 2, we present the Harnack inequality of Krylov–Safonov for divergence elliptic operators in view of some ideas used to study the linearizedMonge–Ampère equation This illustrates these ideas in a case that is simpler thanthat of the linearized Monge–Ampère operator

non-vii

Chapter3presents 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 ofcross sections that are valid under appropriate normalizations using ellipsoids of

minimum volume A typical situation is when the measure Mu satisfies

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 linearoperators The results proved in this chapter permit us to work with the crosssections as if they were Euclidean balls and to establish the covering lemmas neededlater for the regularity theory in Chapters4 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 D1

Chapter5contains Caffarelli’s C1;˛estimates for weak solutions A fundamentalgeometric result is Theorem 5.2.1 about the extremal points of the set where a

solution u equals a supporting hyperplane.

Finally, in Chapter6, we present the W 2;pestimates for the Monge–Ampère tion recently developed by Caffarelli and extend classical estimates of Pogorelov.The main result here is Theorem6.4.2

equa-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, manydiscussions, and for his collaboration I am very grateful to Qingbo Huang forinnumerable enlightening discussions on most topics in this book, for manysuggestions, and corrections, and for his collaboration I am also very grateful toseveral friends and students for carefully reading various chapters of the manuscript:Shif Berhanu, Giuseppe Di Fazio, David Hartenstine, and Federico Tournier Theyhave made many helpful comments, suggestions and corrections that improved thepresentation I would especially like to thank L C Evans for his encouragement andsuggestions

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 tothank 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 bythe National Science Foundation, and I wish to thank this institution for its support

September 2000

Du denotes the gradient of the function u:

D2u x/ denotes the Hessian of the function u, i.e., D2u x/ D

Given a set E,E x/ denotes the characteristic function of E.

jEj denotes the Lebesgue measure of the set E.

B R x/ denotes the Euclidean ball centered at x with radius R.

!ndenotes the measure of the unit ball inRn

C./ denotes the class of real-valued functions that are continuous in :

Given a positive integer k, C k./ denotes the class of real-valued functions thatare continuously differentiable in up to order k.

If E kis a sequence of sets, then

If  Rnis a bounded and measurable set, the center of mass or barycenter of

 is the point xdefined by

xD 1jj

Z

x dx:

xi

xii Notation

If A  B Rnand NA  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 Rnis a cube and˛ > 0, then ˛ Q denotes the cube concentric with Q but

with edge length equals˛ times the edge length of Q.

1 Generalized Solutions to Monge–Ampère Equations 1

1.1 The Normal Mapping 1

1.1.1 Properties of the Normal Mapping 2

1.2 Generalized Solutions 6

1.3 Viscosity Solutions 8

1.4 Maximum Principles 10

1.4.1 Aleksandrov’s Maximum Principle 11

1.4.2 Aleksandrov–Bakelman–Pucci’s Maximum Principle 13

1.4.3 Comparison Principle 17

1.5 The Dirichlet Problem 18

1.6 The Nonhomogeneous Dirichlet Problem 21

1.7 Return to Viscosity Solutions 26

1.8 Ellipsoids of Minimum Volume 28

1.9 Exercises 31

1.10 Notes 39

2 Uniformly Elliptic Equations in Nondivergence Form 41

2.1 Critical Density Estimates 41

2.2 Estimate of the Distribution Function of Solutions 48

2.3 Harnack’s Inequality 51

2.4 Notes 54

3 The Cross-Sections of Monge–Ampère 55

3.1 Introduction 55

3.2 Preliminary Results 57

3.3 Properties of the Sections 63

3.3.1 The Monge–Ampère Measures Satisfying (3.1.1) 63

3.3.2 The Engulfing Property of the Sections 68

3.3.3 The Size of Normalized Sections 70

3.4 Notes 75

xiii

xiv Contents

4 Convex Solutions of det D2uD 1 in Rn

77

4.1 Pogorelov’s Lemma 77

4.2 Interior Hölder Estimates of D2u 81

4.3 C˛Estimates of D2u 84

4.4 Notes 89

5 Regularity Theory for the Monge–Ampère Equation 91

5.1 Extremal Points 91

5.2 A result on extremal points of zeroes of solutions to Monge–Ampère 93

5.3 A Strict Convexity Result 96

5.4 C1;˛Regularity 101

5.4.1 C1;˛Estimates 106

5.5 Examples 111

5.5.1 A Generalization of Formula (5.5.1) 112

5.6 Notes 121

6 W 2;pEstimates for the Monge–Ampère Equation 123

6.1 Approximation Theorem 123

6.2 Tangent Paraboloids 127

6.3 Density Estimates and Power Decay 129

6.4 L pEstimates of Second Derivatives 137

6.5 Proof of the Covering Theorem 6.3.3 141

6.6 Regularity of the Convex Envelope 148

6.7 Notes 150

7 The Linearized Monge–Ampère Equation 153

7.1 Introduction 153

7.2 Normalized Solutions 155

7.3 Critical Density 156

7.4 Double Section Property 161

7.4.1 A1Condition on Sections 163

7.4.2 Behavior of nonnegative Solutions in Expanded Sections 165

7.5 A Calderón-Zygmund Type Decomposition for Sections 172

7.6 Power Decay 181

7.7 Interior Harnack’s Inequality 187

7.8 Notes 191

8 Interior Hölder Estimates for Second Derivatives 193

8.1 Introduction 193

8.2 Interior C2;˛Estimates 193

8.3 Notes 209

Bibliography 211

Index 215

Generalized Solutions to Monge–Ampère

Equations

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.R n / defined by

@u.x0/ D fp W u.x/  u.x0/ C p  x  x0/; for all x 2 g:

Given E  , we define @u.E/ DSx2E @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 Exercise11 If u 2 C2./

and x 2 S, then the Hessian of u is nonnegative definite, that is D2u x/  0: This means that if u is C2, then S is the set where the graph of u is concave up Indeed,

by Taylor’s Theorem u.x C h/ D u.x/ C Du.x/  h C 1

2hD2u ./h; hi, where  lies

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

graph is a cone inRnC1: Let  D B R x0/ in Rn

, h > 0 and u.x/ D h jx  x0j

graph of u, for x 2, is an upside-down right-cone in RnC1with vertex at the point

.x0; 0/ and base on the hyperplane x nC1D 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

8ˆ

<

ˆ:

h R

1.1.1 Properties of the Normal Mapping

Lemma 1.1.3 If  Rn is open, u 2 C./ and K   is compact, then @u.K/ is

compact.

Proof Let fp k g  @u.K/ be a sequence We claim that p k are bounded For each k there exists x k 2 K such that p k 2 @u.x k /; that is u.x/  u.x k / C p k  x  x k/ for all

x 2 : Since K is compact, Kı D fx W dist.x; K/  ıg is compact and contained in

 for all ı sufficiently small, and we may assume by passing if necessary through a

subsequence that x k ! x0 Then x k C ıw 2 Kı, and u.x k C ıw/  u.x k / C ıp k  w for all jwj D 1 and for all k If p k ¤ 0 and w D p k

jp kj, then we get maxKıu x/ 

minK u x/ C ıjp k j; for all k Since u is locally bounded, the claim is proved Hence there exists a convergent subsequence p k m ! p0 We claim that p0 2 @u.K/: We shall prove that p0 2 @u.x0/: We have u.x/  u.x k m / C p k m  x  x k m / for all x 2  and, since u is continuous, by letting m ! 1 we obtain u.x/  u.x0/ C p0 x  x0/

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 x02 , 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 given by u x/ D e jxj2

and K D fx 2Rn W jx i j  1; i D 1; : : : ; ng: The set @u.K/ is

a star-shaped symmetric set around the origin that is not convex, see Figure1.1

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  Cjx  yj for all x; y 2 K:

Proof Since u is convex, u has a supporting hyperplane at any x 2 : Let C D supfjpj W p 2 @u.K/g By Lemma1.1.3, C < 1: If x 2 K, then u.y/  u.x/ C p 

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/  u.x/ 

jpjjy  xj: By reversing the roles of x and y we get the lemma. 

Lemma 1.1.7 If  is open and u is Lipschitz continuous in , then u is

differen-tiable a.e in:

Lemma 1.1.8 If u is convex or concave in , then u is differentiable a.e in :

Proof Follows immediately from Lemmas1.1.6and1.1.7 

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  ! R is the

Lemma 1.1.12 If  is open and u is a continuous function in , then the set of

points inRn 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 R n 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 [kk,wherek kC1are open andk 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 2m That is, if

S m D fp 2 R n W there exist x; y 2 , x ¤ y and p 2 @.uj m /.x/ \ @.uj m /.y/g

we have p 2 S m , i.e., S  [ m S m and we then show that each S mhas measure zero

Let ube the Legendre transform of u By Remark1.1.11and Lemma1.1.8, u

is differentiable a.e Let E D fp W uis not differentiable at pg: We shall show that

fp 2 R n 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 x i  p  u.x i /, i D 1; 2: Also

u.z/  x i  z  u.x i / and so u.z/  u.p/ C x i  z  p/ for all z, i D 1; 2 Hence if

uwere differentiable at p, we would have Du.p/ D x i ; i D 1; 2: This completes

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

is a measure, finite on compacts, that is called the Monge–Ampère measure associated with the function u.

Proof By Lemma1.1.3, the classS contains all compact subsets of  Also, if E m

is any sequence of subsets of, then @u [ m E m/ D [m @u.E m /: Hence, if E m2 S,

mD 1; 2; : : : ; then [m E m 2 S: In particular, we may write  D [m K m with K m

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 Lemma1.1.12, 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 fE ig1

iD1be a sequence of disjoint sets in

S and set @u.E i / D H i We must show that

ˇˇ@u[1

iD1E iˇˇ DX1

iD1

jH ij:

H nDŒH n \ H n1[ H n2[    [ H1/ [ ŒH n n H n1[ H n2[    [ H1/ :Then by Lemma1.1.12, jH n \ H n1[ H n2[    [ H1/j D 0 and we obtain

jH n j D jH n n H n1[ H n2[    [ 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

Mu E/ D

Z

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

gW  ! Rn a C1function in  If S0D fx 2  W det g0.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 D2u x/ > 0g: Indeed, let x1; x2 2 A with Du.x1/ D Du.x2/

By convexity u z/  u.x i / C Du.x i /  z  x i / 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 wecan write

Therefore the integral is zero and the integrand must vanish for0  t  1 Since

x2 2 A, it follows that x2C t.x1 x2/ 2 A for t small Therefore x1D 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 \ S0and E n S0are also Borel sets Hence, by theformula of change of variables and Sard’s Theorem,

Mu E/ D Mu.E \ S0/ C Mu.E n S0/ D

which shows (1.1.4)

Example 1.1.16 If u x/ is the cone of Example1.1.2, then the Monge–Ampère

measure associated with u is Mu D jB h =Rj ıx0; where ıx0 denotes the Dirac delta

at x0:

Definition 1.2.1 Let be a Borel measure defined in , an open and convex subset

ofRn The convex function u 2 C./ is a generalized solution, or Aleksandrov

solution, to the Monge–Ampère equation

det D2uD

if the Monge–Ampère measure Mu associated with u defined by (1.1.1) equals :See Exercise31

The following lemma implies that the notion of generalized solution is closed

under uniform limits That is, if u k are generalized solutions to det D2u D in 

and u k ! u uniformly on compact subsets of , then u is also a generalized solution

(ii) If U is open such that U  , then

@u.U/  lim inf

k!1 @u k U/;

where the inequality holds for almost every point of the set on the left-hand side1, and by Fatou’s lemma

j@u.U/j  lim inf

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/  u k x/j < ı=2 for all x 2 U and for all k  k0 Let

ıkD max

x2U f`.x/  u k x/ C ı=2g:

Since x0 2 U, we have ı k  `.x0/  u k x0/ C ı=2 D u.x0/  u k x0/ C ı=2 >

ı=2 C ı=2 D 0 We have ık D `.x k /  u k x k / C ı=2 for some x k 2 U We claim that x k … @U Otherwise, by definition of ı, `.x k /  u.x k/  ı and

so ık  ı=2, a contradiction We claim that p is the slope of a supporting hyperplane to u kat the point.x k ; u k x k// Indeed,

ık D u.x0/ C p  x k  x0/  u k x k/ C ı=2and so

u k x/  u k x k / C p  x  x k/ 8x 2 U: (1.2.1)

Since u k is convex in , U is open, and x k 2 U, it follows that (1.2.1)

holds for all x 2 , that is p 2 @u k x k / for all k  k0: This implies that

p2 lim infk!1 @u k U/:

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

x1; x22 , x1¤ x2g:

8 1 Generalized Solutions to Monge–Ampère Equations

Finally, we remove the assumption that U is compact If U is open, and

U  , then we can write U DS1jD1U j with U j open and U jcompact Then

Lemma 1.2.3 If u k are convex functions in  such that u k ! u uniformly on

compact subsets of , then the associated Monge–Ampère measures Mu k tend to

for every f continuous with compact support in:

See Exercise21for a proof

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 D2u D f

0/ for all x in a neighborhood of x0, then we must have

det D2 0/  /f x0/:

Remark 1.3.2 We claim that if u 2 C 2

local maximum at x02 , then

Dividing this expression by, letting ! 0 and noting that the resulting inequality

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 convexquadratic polynomials We shall first prove that, if the statement giving a strictly

all x in a neighborhood of x0implies that

has a local maximum at x0 Hence u  P has a local

maximum at x0 Then det D2P .x0/ D detD2 0/ C 2 Id  f x0/ By letting ! 0, we obtain the desired inequality

To prove the statement for supersolutions, let 2./ be convex such that

0/ has some zero eigenvalue, then

det D2 0/ D 0  f x0/: If all eigenvalues of D2

0/ are positive and P.x/ is

given by (1.3.1), then P .x/ D P.x/  jx  x0j2 is strictly convex for all > 0

sufficiently small Proceeding as before, we now get that uP has a local minimum

at x0and consequently det D2 0/  f x0/:

10 1 Generalized Solutions to Monge–Ampère Equations

We now compare the two notions of solutions: generalized solutions andviscosity 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

maximum at x0 2 : We can assume that u.x0 0

0 < jx  x0j  ı This can be achieved by adding rjx  x0j2to

By the continuity of f we obtain that det D2 0/  f x0/:

We shall prove in Section1.7the converse of Proposition1.3.4

Sincev.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 x12  such that a D v.x0/ C p  x1 x0/  u.x1/ and so

and consequently u.x0/ C p  x  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  C nn1dist x0; @/ j@u./j;

for all x02 , where C n is a constant depending only on the dimension n.

Proof Fix x02  and let v be the convex function whose graph is the upside-downcone with vertex.x0; u.x0// and base , with v D 0 on @ Since u is convex, v  u

in By Lemma1.4.1

@v./  @u./:

To prove the theorem, we shall estimate the measure of@v./ from below We firstnotice 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 ofv is a cone,thenv.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

Second, we notice that there exists p0 2 @v./ such that jp0j D u.x0/

dist.x0; @/:This follows because is convex Indeed, we take x1 2 @ such that jx1 x0j Ddist.x0; @/ and let H be the unique supporting hyperplane to the set  at x1 The

uniqueness follows because B jx1x0j.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 tov and has equation z D u.x0/ C p0  x  x0/ with

p0D u x0/

jx0 x1j2.x0 x1/

Now notice that the ball B with center at the origin and radius u.x0/

contained in@v./, and jp0j  u.x0/

 Hence the convex hull of B and p0 iscontained in@v./ and it has measure greater than or equal to

Remark 1.4.3 The estimate in Theorem1.4.2 is meaningful only if j@u./j D

Mu./ < 1: If, for example,

u x/ D

(

px; if0  x  1=2,

p1  x; if 1=2  x  1;

then@u 0; 1// D 1; 1/ We also notice that an estimate of u in terms of the

distance to the boundary of.0; 1/ is not valid

1.4.2 Aleksandrov–Bakelman–Pucci’s Maximum Principle

Consider u 2 C./ with  convex and the classes of functions

F u/ D fv W v.x/  u.x/ 8x 2 ; v convex in g;

G.u/ D fw W w.x/  u.x/ 8x 2 ; w concave in g:

Let

u x/ D sup

v2F u/ v.x/; u.x/ D inf

We have that uis convex and uis concave in We call these functions the convex

and concave envelopes of u in, respectively, and we have the inequalities

Consider the sets of contact points

C.u/ D fx 2  W u.x/ D u.x/g; C.u/ D fx 2  W u.x/ D u.x/g:

Then

14 1 Generalized Solutions to Monge–Ampère Equations

Since u is convex, it follows that u has a supporting hyperplane at x0, for

x0 2 C.u/ Since in addition u.x0/ D u.x0/, this hyperplane is also a supporting

hyperplane to u at the same point That is,

@.u/.x0/  @u.x0/; for x02 C.u/;

and hence

@.u/.C.u//  @u.C.u//:

If x0 … C.u/, then @u.x0/ D ;: Also, if A; B are sets, then @u.A [ B/ D @u.A/ [

Notice thatˆu x0/ D @u.x0/:

Lemma 1.4.4 Let u 2 C./ such that u.x/  0 on @, and x02  with u.x0/ > 0.

Consider the minimum

and0would not be the infimum

We claim that 2  Since uj@  0, the claim will be proved if we show

u / > 0 By taking  D x0in (1.4.5) we get u.x0/  0, and consequently

u /  y    x0/ C 0; 8 2 :

This means that0 C y    x0/ is a supporting hyperplane of u at  Since u

is minimal, we have u./  u./  y    x0/ C 0; 8 2 : In particular,

u / D u./ and y    x0/ C 0 is a supporting hyperplane for uat , i.e.,

Then under the assumptions of Lemma1.4.4we get

.x0; u.x0//  ˆu.C.u// D @..u//.C.u// D @ u//.C.u//:

We also have the estimate

j.x0; t/j  !n t n

16 1 Generalized Solutions to Monge–Ampère Equations

To show (1.4.6), we first note that

.x0; t/ D t.x0; 1/;

which follows by writing y    x0/ C t D t y

t    x0/ C 1: Also, if x0 2 ,then

.diam.//n  j  @ u//.C.u//j:

We then obtain the following maximum principle:

Theorem 1.4.5 (Aleksandrov–Bakelman–Pucci’s Maximum Principle) If u2

C / and u  0 on @, then

max

 u x/  ! 1=n

n diam / j@ u//.C.u//j 1=n:

If in addition u 2 C2./ (without any assumptions on the sign of u on @), then

Proof It only remains to prove the last inequality Subtracting from u the maximum

on the boundary, we may assume that u 0 on @: From (1.4.2), (1.4.3), and (1.4.4)

we get@ u// C.u// D @.u/ C.u// : If u 2 C2 and z 2 C.u/, then

D2.u/.z/  0: Thus, by the formula for change of variables we obtain

1.4.3 Comparison Principle

Theorem 1.4.6 Let u ; v 2 C./, with v convex, such that

j@u.E/j  j@v.E/j; for every Borel set E :

Then

min

x2fu.x/  v.x/g D min

x2@fu.x/  v.x/g:

Proof The proof is by contradiction Let a D min x2fu.x/  v.x/g, b D

minx2@fu.x/  v.x/g, and assume a < b: There exists x0 2  such that a D

u x0/  v.x0/: Pick ı > 0 sufficiently small such that ı diam/2< b  a2 , and let

w x/ D v.x/ C ıjx  x0j2C b C a

Consider the set G D fx 2  W u.x/ < w.x/g: We have x02 G Also, G \ @ D ;.

In fact, if x 2 G \ @, then u.x/  v.x/  b and so

Hence w.x/ < u.x/ for x 2 @: This implies that @G D fx 2  W u.x/ D w.x/g: By

Lemma1.4.1we obtain@w.G/  @u.G/ Also @w D @.v C ıjx  x0j2/; and fromLemma1.4.7below we have the inequality

j@.v C ıjx  x0j2/.G/j  j@v.G/j C j@.ıjx  x0j2/.G/j: (1.4.8)Therefore

j@u.G/j  j@w.G/j  j@v.G/j C j@.ıjx  x0j2/.G/j D j@v.G/j C 2ı/ n jGj;

Lemma 1.4.7 If

M for each Borel set E  .

18 1 Generalized Solutions to Monge–Ampère Equations

Let ... because it has corners on the line x D1:

In this section we solve the nonhomogeneous Dirichlet problem for the Monge? ? ?Ampère operator using the Perron method and Theorem1.5.2 Let be an... D be the equation of the supporting hyperplane to  at the point , and

The second. .. @u./:

To prove the theorem, we shall estimate the measure of@v./ from below We firstnotice that the set@v./ is convex This is true because, if p @v./, then there exists x1