Convex Analysis and Nonlinear Geometric Elliptic Equations

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.