Convex Analysis and Nonlinear Geometric Elliptic Equations

I. Bakelman was an expert in the study of nonlinear elliptic partial differential equations by methods of differential and convex geometry. In Russia he is also recognized as a reformer of mathematical education at both school and university levels. This book represents much of Bakelman's work of the last ten years (until his death in 1992). Much of his work was devoted to boundary value problems for mean curvature and Monge-Ampere equations in more than two variables and their generalizations. The book is suitable as a text book and reference work for graduate students and scientists (mathematicians but also physicists) working in the areas of convex functions and bodies, global geometric problems and nonlinear elliptic boundary value problems.

I. Elements of Convex Analysis.- 1. Convex Bodies and Hypersurfaces.- §1. Convex Sets in Finite-Dimensional Euclidean Spaces.- 1.1. Main Definition.- 1.2. Linear and Convex Operations with Convex Sets. Convex Hull.- 1.3. The Properties of Convex Sets in Linear Topological Spaces.- 1.4. Euclidean Space En.- 1.5. The Simple Figures in En.- 1.6. Spherical Convex Sets.- 1.7. Starshapedness of Convex Bodies.- 1.8. Asymptotic Cone.- 1.9. Complete Convex Hypersurfaces in En+1.- §2. Supporting Hyperplanes.- 2.1. Supporting Hyperplanes. The Separability Theorem.- 2.2. The Main Properties of Supporting Hyperplanes.- §3. Convex Hypersurfaces and Convex Functions.- 3.1. Convex Hypersurfaces and Convex Functions.- 3.2. Test of Convexity of Smooth Functions.- 3.3. Convergence of Convex Functions.- 3.4. Convergence in Topological Spaces.- 3.5. Convergence of Convex Bodies and Convex Hypersurfaces.- §4. Convex Polyhedra.- 4.1. Definitions. Description of Convex Polyhedra by the Convex Hull of Their Vertices.- 4.2. Convex Hull of a Finite System of Points.- 4.3. Approximation of Closed Convex Hypersurfaces by Closed Convex Polyhedra.- §5. Integral Gaussian Curvature.- 5.1. Spherical Mapping and the Integral Gaussian Curvature.- 5.2. The Convergence of Integral Gaussian Curvatures.- 5.3. Infinite Convex Hypersurfaces.- §6. Supporting Function.- 6.1. Definition and Main Properties.- 6.2. Differential Geometry of Supporting Function.- 2. Mixed Volumes. Minkowski Problem. Selected Global Problems in Geometric Partial Differential Equations.- §7. The Minkowski Mixed Volumes.- 7.1. Linear Combinations of Sets in En+l.- 7.2. Exercises and Problems to Subsection 7.1.- 7.3. Minkowski Mixed Volumes for Convex Polyhedra.- 7.4. The Minkowski Mixed Volumes for General Bounded Convex Bodies.- 7.5. The Brunn-Minkowski Theorem. The Minkowski Inequalities.- 7.6. Alexandrov’s and Fenchel’s Inequalities.- §8. Selected Global Problems in Geometric Partial Differential Equations.- 8.1. Minkowski’s Problem for Convex Polyhedra in En+1.- 8.2. The Classical Minkowski Theorem.- 8.3. General Elliptic Operators and Equations.- 8.4. Linear Elliptic Operators and Equations.- 8.5. Quasilinear Elliptic Operators and Equations.- 8.6. The Classical Monge-Ampere Equations.- 8.7. Differential Equations in Global Problems of Differential Geometry.- 8.8. The Classical Maximum Principles for General Elliptic Equations.- 8.9. Hopf’s Maximum Principle for Uniformly Elliptic Linear Equations.- 8.10. Uniqueness Theorem for General Nonlinear Elliptic Equations.- 8.11. The Maximum Principle for Divergent Quasilinear Elliptic Equations.- 8.12. Uniqueness Theorem for Isometric Embeddings of Two-dimensional Riemannian Metrics in E3.- II. Geometric Theory of Elliptic Solutions of Monge-Ampere Equations.- 3. Generalized Solutions of N-Dimensional Monge-Ampere Equations.- §9. Normal Mapping and R-Curvature of Convex Functions.- 9.1. Some Notation.- 9.2. Normal Mapping.- 9.3. Convergence Lemma of Supporting Hyperplanes.- 9.4. Main Properties of the Normal Mapping of a Convex Hypersurface.- 9.5. Proofs.- 9.6. R -curvature of convex functions.- 9.7. Weak convergence of R-curvatures.- §10. The Properties of Convex Functions Connected with Their R-Curvature.- 10.1. The Comparison and Uniqueness Theorems.- 10.2. Geometric Lemmas and Estimates.- 10.3. The Border of a Convex Function.- 10.4. Convergence of Convex Functions in a Closed Convex Domain. Compactness Theorems.- §11. Geometric Theory of the Monge-Ampere Equations det (uij) = ?(x)/R(Du).- 11.1. Introduction. Obstructions and Necessary Conditions of Solvability for the Dirichlet Problem.- 11.2. Generalized and Weak Solutions for Equation (11.1).- 11.3. The Dirichlet Problem in the Set of Convex Functions Q(A1,A2,...,Ak).- 11.4. Existence and Uniqueness of Weak Solutions of the Dirichlet Problem for Monge-Ampere Equations det(uij) = ?( x)/R (Du).- 11.5. The Inverse Operator for the Dirichlet Problem.- 11.6. Hypersurfaces with Prescribed Gaussian Curvature.- §12. The Dirichlet Problem for Elliptic Solutions of Monge-Ampere Equations Det(uij) = f(x,u, Du).- 12.1. The First Main Existence Theorem for the Dirichlet Problem (12.1-2).- 12.2. Existence of at Least One Generalized Solution of the Dirichlet Problem for Equations det (uij) = f (x,u,Du).- 12.3. Existence of Several Different Generalized Solutions for the Dirichlet Problem (12.23-24).- 4. Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations.- §13. Introduction. The Main Functional.- 13.1. Statement of Problems.- 13.2. Preliminary Considerations.- 13.3. The Functional IH(u) and its Properties.- §14. Variational Problem for the Functional IH(u).- 14.1. Bilateral Estimates for IH (u).- 14.2. Main Theorem about the Functional IH(u).- §15. Dual Convex Hypersurfaces and Euler’s Equation.- 15.1. Special Map on the Hemisphere.- 15.2. Dual Convex Hypersurfaces.- 15.3. Expression of the Functional IH(u)by Means of Dual Convex Hypersurfaces.- 15.4. Expression of the Variation of IH(u).- 5. Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations.- §16. Introduction. The Statement of the Second Boundary Value Problem.- 16.1. Asymptotic Cone of Infinite Complete Convex Hypersurfaces.- 16.2. The Statement of the Second Boundary Value Problem.- §17. The Second Boundary Value Problem for Monge-Ampere Equations det $$\det \left( {{u_{{ij}}}} \right) = \frac{{g\left( x \right)}}{{R\left( {Du} \right)}} $$.- 17.1. The Necessary and Sufficient Conditions of Solvability of the Second Boundary Value Problem.- 17.2. The Second Boundary Value Problem in the Class of Convex Polyhedra.- §18. The Second Boundary Value Problem for General Monge-Ampere Equations.- 18.1. The Main Assumptions.- 18.2. The Statement of the Main Theorem and the Scheme of its Proof.- 18.3. The Function Space of the Second Boundary Value Problem.- 18.4. The Proof of Theorem 18.1.- 6. Smooth Elliptic Solutions of Monge-Ampere Equations.- §19. The N-Dimensional Minkowski Problem.- 19.1. Introduction.- 19.2. A Priori Estimates for the Radii of Normal Curvature of a Convex Hypersurface.- 19.3. Auxiliary Concepts and Formulas Obtained by E. Calabi [1] and A. Pogorelov [3].- 19.4. An A Priori Estimate for the Third Derivatives of a Support Function of a Convex Hypersurface.- 19.5. The Proof of Theorem 19.3.- §20. The Dirichlet Problem for Smooth Elliptic Solutions of N-Dimensional Monge-Ampere Equations.- 20.1. The Uniqueness and Comparison Theorems.- 20.2. C°-Estimates for Solutions $$u\left( x \right) \in {C^{2}}\left( {\overline G } \right) $$ of the Dirichlet Problem (20.2) by Subsolutions.- 20.3. Geometric Estimates of Convex Solutions for Monge-Ampere Equations.- 20.4. Geometric Estimates of the Gradient of Convex Solutions for Monge-Ampere Equations.- 20.5. The Dirichlet Problem for the Monge-Ampere Equation det(uij) = ? ( x).- 20.6. A Priori Estimates for Derivatives up to Second Order.- 20.7. Calabi’s Interior Estimates for the Third Derivatives.- 20.8. One-Sided Estimates at the Boundary for some Third Derivatives.- 20.9. An Important Lemma.- 20.10. Completion of the Proof of Theorem 20.8.- 20.11. More General Monge-Ampere equations.- III. Geometric Methods in Elliptic Equations of Second Order. Applications to Calculus of Variations, Differential Geometry and Applied Mathematics..- 7. Geometric Concepts and Methods in Nonlinear Elliptic Euler-Lagrange Equations.- §21. Geometric Constructions. Two-Sided C°-Estimates of Functions with Prescribed Dirichlet Data.- 21.1. Geometric Constructions.- 21.2. Convex and Concave Supports of Functions $$u\left( x \right) \in W_{2}^{n}\left( B \right) \cap C\left( {\overline B } \right) $$.- 21.3. Two-sided C0-Estimates for Functions $$u\left( x \right) \in W_{2}^{n}\left( B \right) \cap C\left( {\overline B } \right) $$.- §22. Applications to the Dirichlet Problem for Euler-Lagrange Equations.- §23. Applications to Calculus of Variations, Differential Geometry and Continuum Mechanics.- 23.1. Applications to Calculus of Variations.- 23.2. Applications to Differential Geometry.- 23.3. Applications to Continuum Mechanics.- §24. C2 -Estimates for Solutions of General Euler-Lagrange Elliptic Equations.- 24.1. Introduction.- 24.2. Monge-Ampere Generators.- 24.3. Assumptions Related to General Euler-Lagrange Equations.- 24.4. Two-sided Estimates for Solutions of Nonlinear Elliptic Euler-Lagrange Equations.- 24.5. The Second Type of C0-Estimates for Solutions for General Elliptic Euler-Lagrange Equations.- 8. The Geometric Maximum Principle for General Non-Divergent Quasilinear Elliptic Equations.- §25. The First Geometrie Maximum Principle for Solutions of the Dirichlet Problem for General Quasilinear Equations.- 25.1. The First Geometrie Maximum Principle for General Quasilinear Elliptic Equations and Linear Elliptic Equations of the Form.- 25.2. The Improvement of Estimates (25.16) for Solutions of General Quasilinear Elliptic Equations Depending on Properties of the Functions det(aik(x,u,p)) and b (x,u,p).- 25.3. The Improvement of Estimate (25.106-107) and (25.113) for Solutions $$u\left( x \right) \in W_{2}^{n}\left( B \right) \cap C\left( {\overline B } \right) $$ of the Dirichlet Problem for Euler-Lagrange Equations.- 25.4. Final Remarks Relating to Subsections 25.2 and 25.3.- 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(CoB).- §26. The Geometric Maximum Principle for General Quasilinear Elliptic Equations (Continuation and Development).- 26.1. The Main Assumptions.- 26.2. Concepts and Notations Related to Solutions of the Dirichlet Problem (26.1-2).- 26.3. The Development of Techniques Related to Functions Q? (p) and R(p).- 26.4. The Main Estimates for Solutions of Problem (26.1-2) if R(p) Satisfies (26.9-a).- 26.5. Uniform Estimates for Solutions of the Dirichlet Problem (26.1-2) (Continuation and Development of Subsection (26.4).- 26.6. Comments to the Modified Condition C.2.- §27. Pointwise Estimates for Solutions of the Dirichlet Problem for General Quasilinear Elliptic Equations.- 27.1. Integral I (?, ?,xo).- 27.2. The Mapping’s Mean.- 27.3. The General Lemma of Convexity.- 27.4. The Pointwise Estimates for Solutions of the Dirichlet Problem (27.1-2).- §28. Comments to Chapter 8. The Maximum Principles in Global Problems of Differential Geometry.- 28.1. Comments to Chapter 8.- 28.2. Estimates for Solutions of Quasilinear Elliptic Equations Connected with Problems of Global Geometry.- §29. The Dirichlet Problem for Quasilinear Elliptic Equations.- 29.1. Introduction.- 29.2. Estimates for the Gradient on the Boundary of ?B. (The Method of Global Barriers).- 29.3. Estimates of the Gradient of Solutions on the Boundary. (The Method of Convex Majorants).- 29.4. Estimates of the Gradient of Solutions on the Boundary. (The Method of Support Hyperplanes).- 29.5. Global Gradient Estimates for Solutions of Quasilinear Elliptic Equations.