عنوان

Convex analysis and nonlinear geometric elliptic equations.

(Number (ISBN

3642698816

(Number (ISBN

9783642698811

Number

b572865

Title Proper

Convex analysis and nonlinear geometric elliptic equations.

General Material Designation

[Book]

First Statement of Responsibility

Ilya J Bakelman

Place of Publication, Distribution, etc.

[Place of publication not identified]

Name of Publisher, Distributor, etc.

Springer

Date of Publication, Distribution, etc.

2012

Text of Note

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

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Ilya J Bakelman

Ilya J Bakelman

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