Ilya J. Bakelman

Convex Analysis and Nonlinear Geometrie Elliptic Equations

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

Contents

Part I. Elements of Convex Analysis 1 Chapter 1. Convex Bodies and Hypersurfaces 3

§!. Convex Sets in Finite-Dimensional Euclidean Spaces 3 1.1. Main Definition 3 1.2. Linear and Convex Operations with Convex Sets.

Convex Hüll 3 1.3. The Properties of Convex Sets in Linear Topological Spaces 7 1.4. Euclidean Space En 8 1.5. The Simple Figures in En 9 1.6. Spherical Convex Sets 10 1.7. Starshapedness of Convex Bodies 11 1.8. Asymptotic Cone 12 1.9. Complete Convex Hypersurfaces in En+1 13

$2. Supporting Hyperplanes 14 2.1. Supporting Hyperplanes. The Separability Theorem 14 2.2. The Main Properties of Supporting Hyperplanes 15

§5. Convex Hypersurfaces and Convex Functions 16 3.1. Convex Hypersurfaces and Convex Functions 16 3.2. Test of Convexity of Smooth Functions 19 3.3. Convergence of Convex Functions 20 3.4. Convergence in Topological Spaces 21 3.5. Convergence of Convex Bodies and Convex Hypersurfaces 22

y . Convex Polyhedra 24 4.1. Definitions. Description of Convex Polyhedra by the Convex Hüll

of Their Vertices 24 4.2. Convex HuU of a Finite System of Points 26 4.3. Approximation of Closed Convex Hypersurfaces

by Closed Convex Polyhedra 28

§5. Integral Gaussian Curvature 29 5.1. Spherical Mapping and the Integral Gaussian Curvature 29 5.2. The Convergence of Integral Gaussian Curvatures 33 5.3. Infinite Convex Hypersurfaces 35

§£. Supporting Function 36 6.1. Definition and Main Properties 36 6.2. Differential Geometry of Supporting Function 41

Chapter 2. Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometrie Partial Differential Equations 54

XVI Contents

§7. The Minkowski Mixed Volumes 54 7.1. Linear Combinations of Sets in E"+1 54 7.2. Exercises and Problems to Subsection 7.1 58 7.3. Minkowski Mixed Volumes for Convex Polyhedra 59 7.4. The Minkowski Mixed Volumes

for General Bounded Convex Bodies 63 7.5. The Brunn-Minkowski Theorem.

The Minkowski Inequalities 66 7.6. Alexandrov's and FenchePs Inequalities 72

§8. Selected Global Problems in Geometrie Partial Differential Equations 75

8.1. Minkowski's Problem for Convex Polyhedra in £ ^ + 1 75 8.2. The Classical Minkowski Theorem 80 8.3. General Elliptic Operators and Equations 86 8.4. Linear Elliptic Operators and Equations 87 8.5. Quasilinear Elliptic Operators and Equations 88 8.6. The Classical Monge-Ampere Equations 89 8.7. Differential Equations in Global Problems

of Differential Geometry 90 8.8. The Classical Maximum Principles

for General Elliptic Equations 95 8.9. Hopfs Maximum Principle for Uniformly Elliptic

Linear Equations 98 8.10. Uniqueness Theorem

for General Nonlinear Elliptic Equations 100 8.11. The Maximum Principle

for Divergent Quasilinear Elliptic Equations 103 8.12. Uniqueness Theorem for Isometric Embeddings

of Two-dimensional Riemannian Metrics in E3 105

Part IL Geometrie Theory of Elliptic Solutions of Monge-Ampere Equations 109 C h a p t e r 3. General ized Solut ions of iV-Dimensional M o n g e - A m p e r e Equa t i ons 113

§5. Normal Mapping and R-Curvature of Convex Functions 113 9.1. Some Notation 113 9.2. Normal Mapping 113 9.3. Convergence Lemma of Supporting Hyperplanes 114 9.4. Main Properties of the Normal Mapping

of a Convex Hypersurface 115 9.5. Proofs 116 9.6. .R-curvature of convex funetions 118 9.7. Weak convergence of iZ-curvatures 118

Contents XVII

§10. The Properties of Convex Functions Connected urith Their R-Curvature 123

10.1. The Comparison and Uniqueness Theorems 123 10.2. Geometrie Lemmas and Estimates 125 10.3. The Border of a Convex Function 127 10.4. Convergence of Convex Functions in a Closed Convex Domain.

Compactness Theorems 129

§11. Geometrie Theory of the Monge-Ampere Equations Aet{uij) = <p(x)/R(Du) 146

11.1. Introduction. Obstructions and Necessary Conditions of Solvability for the Dirichlet Problem 146

11.2. Generalized and Weak Solutions for Equation (11.1) 148 11.3. The Dirichlet Problem in the Set of Convex Functions

Q(AuA2,...,Ak) 150 11.4. Existence and Uniqueness of Weak Solutions

of the Dirichlet Problem for Monge-Ampere Equations det(uij) = <p(x)/R(Du) 153

11.5. The Inverse Operator for the Dirichlet Problem 159 11.6. Hypersurfaces with Prescribed Gaussian Curvature 161

§12. The Dirichlet Problem for Elliptic Solutions of Monge-Ampere Equations Det(ujj) = f(x, u, Du) 166

12.1. The First Main Existence Theorem for the Dirichlet Problem (12.1-2) 167

12.2. Existence of at Least One Generalized Solution of the Dirichlet Problem for Equations det(tiij) = f(x,u,Du) 170

12.3. Existence of Several Different Generalized Solutions for the Dirichlet Problem (12.23-24) 173

Chapter 4. Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations 182

§13. Introduction. The Main Punctional 182 13.1. Statement of Problems 182 13.2. Preliminary Considerations 183 13.3. The Functional In (u) and its Properties 184

§14. Variational Problem for the Functional /// (u) 189 14.1. Bilateral Estimates for IH(U) 189 14.2. Main Theorem about the Functional J# (u) 192

§15. Dual Convex Hypersurfaces and Euler's Equation 193 15.1. Special Map on the Hemisphere 194 15.2. Dual Convex Hypersurfaces 194

XVIII Contents

15.3. Expression of the Functional IH (U) by Means of Dual Convex Hypersurfaces 198

15.4. Expression of the Variation of IH(U) 200

Chapter 5. Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations 204

$16. Introduction. The Statement of the Second Boundary Value Problem 204

16.1. Asymptotic Cone of Infinite Complete Convex Hypersurfaces 204

16.2. The Statement of the Second Boundary Value Problem 205

$17. The Second Boundary Value Problem for Monge-Ampere Equations det(ujj) = nföL 207

17.1. The Necessary and Sufficient Conditions of Solvability of the Second Boundary Value Problem 207

17.2. The Second Boundary Value Problem in the Class of Convex Polyhedra 208

$18. The Second Boundary Value Problem for General Monge-Ampere Equations 212

18.1. The Main Assumptions 212 18.2. The Statement of the Main Theorem

and the Scheme of its Proof. 213 18.3. The Function Space

of the Second Boundary Value Problem 214 18.4. The Proof of Theorem 18.1 218

Chapter 6. Smooth Elliptic Solutions of Monge-Ampere Equations 226

$19. The N-Dimensional Minkowski Problem 226 19.1. Introduction 226 19.2. A Priori Estimates for the Radii of Normal Curvature

of a Convex Hypersurface 228 19.3. Auxiliary Concepts and Formulas Obtained

by E. Calabi [1] and A. Pogorelov [3] 231 19.4. An A Priori Estimate for the Third Derivatives

of a Support Function of a Convex Hypersurface 235 19.5. The Proof of Theorem 19.3 238

$20. The Dirichlet Problem for Smooth Elliptic Solutions of N-Dimensional Monge-Ampere Equations 241

20.1. The Uniqueness and Comparison Theorems 242 20.2. C°-Estimates for Solutions u{x) G C2(G)

of the Dirichlet Problem (20.2) by Subsolutions 244 20.3. Geometrie Estimates of Convex Solutions

for Monge-Ampere Equations 252

Contents XIX

20.4. Geometrie Estimates of the Gradient of Convex Solutions for Monge-Ampere Equations 260

20.5. The Dirichlet Problem for the Monge-Ampere Equation det(uij) = ip(x) 264

20.6. A Priori Estimates for Derivatives up to Second Order 266 20.7. Calabi's Interior Estimates for the Third Derivatives 271 20.8. One-Sided Estimates at the Boundary

for some Third Derivatives 275 20.9. An Important Lemma 277

20.10. Completion of the Proof of Theorem 20.8 280 20.11. More General Monge-Ampere equations 283

Part III. Geometrie Methods in Elliptic Equations of Second Order. Applications to Calculus of Variations, Differential Geometry and Applied Mathematics. 285 Chapter 7. Geomet r i e Concep t s and M e t h o d s in Nonlinear Ell iptic Eu le r -Lag range Equa t ions 287

$21. Geometrie Constructions. Two-Sided C°-Estimates of Functions with Prescribed Dirichlet Data 288

21.1. Geometrie Constructions 288 21.2. Convex and Concave Supports

of Functions u{x) G W%(B) n C(B) . ^ . . . . 289 21.3. Two-sided C°-Estimates for Functions u(x) e W£(B) n C(B) . . . 290

§22. Applications to the Dirichlet Problem for Euler-Lagrange Equations 297

§23. Applications to Calculus of Variations, Differential Geometry and Continuum Mechanics 303

23.1. Applications to Calculus of Variations 303 23.2. Applications to Differential Geometry 306 23.3. Applications to Continuum Mechanics 308

§24- C2-Estimates for Solutions of General Euler-Lagrange Elliptic Equations 312

24.1. Introduction 312 24.2. Monge-Ampere Generators 313 24.3. Assumptions Related to General Euler-Lagrange Equations 323 24.4. Two-sided Estimates for Solutions

of Nonlinear Elliptic Euler-Lagrange Equations 328 24.5. The Second Type of C°-Estimates for Solutions

for General Elliptic Euler-Lagrange Equations 330

Chapter 8. The Geomet r i e Maximum Principle for General Non-Divergent Quasilinear Elliptic Equations 339

XX Contents

$25. The First Geometrie Maximum Principle for Solutions of the Dirichlet Problem for General Quasilinear Equations 341

25.1. The First Geometrie Maximum Principle for General Quasilinear Elliptic Equations and Linear Elliptic Equations of the Form 341

25.2. The Improvement of Estimates (25.16) for Solutions of General Quasilinear Elliptic Equations Depending on Properties of the Functions det(aifc(x,u,p)) and b(x,u,p) 362

25.3. The Improvement of Estimate (25^106-107) and (25.113) for Solutions u{x) e W£(B) n C{B) of the Dirichlet Problem for Euler-Lagrange Equations 377

25.4. Final Remarks Relating to Subsections 25.2 and 25.3 380 25.5. Polar Reciprocal Convex Bodies. Estimates and Majorants

for Solutions of the Dirichlet Problems (25.119-120) and (25.186-187) depending on vol(Coö) 380

$26. The Geometrie Maximum Principle for General Quasilinear Elliptic Equations (Continuation and Development) 384

26.1. The Main Assumptions 385 26.2. Concepts and Notations Related to Solutions

of the Dirichlet Problem (26.1-2) 386 26.3. The Development of Techniques Related

to Functions QQ(|p|) and R(\p\) 388 26.4. The Main Estimates for Solutions of Problem (26.1-2)

if R(p) Satisfies (26.9-a) 390 26.5. Uniform Estimates for Solutions

of the Dirichlet Problem (26.1-2) (Continuation and Development of Subsection (26.4) 395

26.6. Comments to the Modified Condition C.2 402

$27. Pointwise Estimates for Solutions of the Dirichlet Problem for General Quasilinear Elliptic Equations 405

27.1. Integral I(X,a,x0) 405 27.2. The Mapping's Mean 407 27.3. The General Lemma of Convexity 409 27.4. The Pointwise Estimates for Solutions

of the Dirichlet Problem (27.1-2) 413

$28. Comments to Chapter 8. The Maximum Principles in Global Problems of Differential Geometry 441

28.1. Comments to Chapter 8 441 28.2. Estimates for Solutions

of Quasilinear Elliptic Equations Connected with Problems of Global Geometry 442

Contents XXI

$29. The Dirichlet Problem for Quasilinear Elliptic Equations 446 29.1. Introduction 446 29.2. Estimates for the Gradient on the Boundary

of dB. (The Method of Global Barriers) 450 29.3. Estimates of the Gradient

of Solutions on the Boundary. (The Method of Convex Majorants) 465

29.4. Estimates of the Gradient of Solutions on the Boundary. (The Method of Support Hyperplanes) 473

29.5. Global Gradient Estimates for Solutions of Quasilinear Elliptic Equations 488

Bibliography 497

Index 509