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  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
Band 252

Nonlinear Analysis on Manifolds. Monge-Ampère Equations

Aus der Reihe Grundlehren der mathematischen Wissenschaften

Fr. 191.00

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

15.12.1982

Verlag

Springer Us

Seitenzahl

204

Maße (L/B/H)

24.1/16/1.7 cm

Gewicht

490 g

Auflage

1982

Sprache

Englisch

ISBN

978-0-387-90704-8

Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

15.12.1982

Verlag

Springer Us

Seitenzahl

204

Maße (L/B/H)

24.1/16/1.7 cm

Gewicht

490 g

Auflage

1982

Sprache

Englisch

ISBN

978-0-387-90704-8

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: GPSR Kontakt

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  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
  • Produktbild: Nonlinear Analysis on Manifolds. Monge-Ampère Equations
  • 1 Riemannian Geometry.-
    1. Introduction to Differential Geometry.- 1.1 Tangent Space.- 1.2 Connection.- 1.3 Curvature.-
    2. Riemannian Manifold.- 2.1 Metric Space.- 2.2 Riemannian Connection.- 2.3 Sectional Curvature. Ricci Tensor. Scalar Curvature.- 2.4 Parallel Displacement. Geodesic.-
    3. Exponential Mapping.-
    4. The Hopf-Rinow Theorem.-
    5. Second Variation of the Length Integral.- 5.1 Existence of Tubular Neighborhoods.- 5.2 Second Variation of the Length Integral.- 5.3 Myers’ Theorem.-
    6. Jacobi Field.-
    7. The Index Inequality.-
    8. Estimates on the Components of the Metric Tensor.-
    9. Integration over Riemannian Manifolds.-
    10. Manifold with Boundary.- 10.1. Stokes’ Formula.-
    11. Harmonic Forms.- 11.1. Oriented Volume Element.- 11.2. Laplacian.- 11.3. Hodge Decomposition Theorem.- 11.4. Spectrum.- 2 Sobolev Spaces.-
    1. First Definitions.-
    2. Density Problems.-
    3. Sobolev Imbedding Theorem.-
    4. Sobolev’s Proof.-
    5. Proof by Gagliardo and Nirenberg.-
    6. New Proof.-
    7. Sobolev Imbedding Theorem for Riemannian Manifolds.-
    8. Optimal Inequalities.-
    9. Sobolev’s Theorem for Compact Riemannian Manifolds with Boundary.-
    10. The Kondrakov Theorem.-
    11. Kondrakov’s Theorem for Riemannian Manifolds.-
    12. Examples.-
    13. Improvement of the Best Constants.-
    14. The Case of the Sphere.-
    15. The Exceptional Case of the Sobolev Imbedding Theorem.-
    16. Moser’s Results.-
    17. The Case of the Riemannian Manifolds.-
    18. Problems of Traces.- 3 Background Material.-
    1. Differential Calculus.- 1.1. The Mean Value Theorem.- 1.2. Inverse Function Theorem.- 1.3. Cauchy’s Theorem.-
    2. Four Basic Theorems of Functional Analysis.- 2.1. Hahn-Banach Theorem.- 2.2. Open Mapping Theorem.- 2.3. The Banach-Steinhaus Theorem.- 2.4. Ascoli’s Theorem.-
    3. Weak Convergence. Compact Operators.- 3.1. Banach’s Theorem.- 3.2. The Leray-Schauder Theorem.- 3.3. The Fredholm Theorem.-
    4. The Lebesgue Integral.- 4.1. Dominated Convergence Theorem.- 4.2. Fatou’s Theorem.- 4.3. The Second Lebesgue Theorem.- 4.4. Rademacher’s Theorem.- 4.5. Fubini’s Theorem.-
    5. The LpSpaces.- 5.1. Regularization.- 5.2. Radon’s Theorem.-
    6. Elliptic Differential Operators.- 6.1. Weak Solution.- 6.2. Regularity Theorems.- 6.3. The Schauder Interior Estimates.-
    7. Inequalities.- 7.1. Hölder’s Inequality.- 7.2. Clarkson’s Inequalities.- 7.3. Convolution Product.- 7.4. The Calderon-Zygmund Inequality.- 7.5. Korn-Lichtenstein Theorem.- 7.6. Interpolation Inequalities.-
    8. Maximum Principle.- 8.1. Hopf’s Maximum Principle.- 8.2. Uniqueness Theorem.- 8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two.- 8.4. Generalized Maximum Principle.-
    9. Best Constants.- 9.1. Application to Sobolev Spaces.- 4 Green’s Function for Riemannian Manifolds.-
    1. Linear Elliptic Equations.- 1.1. First Nonzero Eigenvalue ? of ?.- 1.2. Existence Theorem for the Equation ?? = f.-
    2. Green’s Function of the Laplacian.- 2.1. Parametrix.- 2.2. Green’s Formula.- 2.3. Green’s Function for Compact Manifolds.- 2.4. Green’s Function for Compact Manifolds with Boundary.- 5 The Methods.-
    1. Yamabe’s Equation.- 1.1. Yamabe’s Method.-
    2. Berger’s Problem.- 2.1. The Positive Case.-
    3. Nirenberg’s Problem.- 3.1. A Nonlinear Theorem of Fredholm.- 3.2. Open Questions.- 6 The Scalar Curvature.-
    1. The Yamabe Problem.- 1.1. Yamabe’s Functional.- 1.2. Yamabe’s Theorem.-
    2. The Positive Case.- 2.1. Geometrical Applications.- 2.2. Open Questions.-
    3. Other Problems.- 3.1. Topological Meaning of the Scalar Curvature.- 3.2. Kazdan and Warner’s Problem.- 7 Complex Monge-Ampere Equation on Compact Kähler Manifolds.-
    1. Kähler Manifolds.- 1.1 First Chern Class.- 1.2. Change of Kahler Metrics. Admissible Functions.-
    2. Calabi’s Conjecture.-
    3. Einstein-Kähler Metrics.-
    4. Complex Monge-Ampere Equation.- 4.1. About Regularity.- 4.2. About Uniqueness.-
    5. Theorem of Existence (the Negative Case).-
    6. Existence of Kähler-Einstein Metric.-
    7. Theorem of Existence (the Null Case).-
    8. Proof of Calabi’s Conjecture.-
    9. The Positive Case.-
    10. A Priori Estimate for ??.-
    11. A Priori Estimate for the Third Derivatives of Mixed Type.-
    12. The Method of Lower and Upper Solutions.- 8 Monge-Ampère Equations.-
    1. Monge-Ampère Equations on Bounded Domains of ?n.- 1.1. The Fundamental Hypothesis.- 1.2. Extra Hypothesis.- 1.3. Theorem of Existence.-
    2. The Estimates.- 2.1. The First Estimates.- 2.2. C2-Estimate.- 2.3. C3-Estimate.-
    3. The Radon Measure ?(?).-
    4. The Functional ? (?).- 4.1. Properties of ? (?).-
    5. Variational Problem.-
    6. The Complex Monge-Ampère Equation.- 6.1. Bedford’s and Taylor’s Results.- 6.2. The Measure M(?).- 6.3. The Functional J(?).- 6.4. Some Properties of J(?).-
    7. The Case of Radially Symmetric Functions.- 7.1. Variational Problem.- 7.2. An Open Problem.-
    8. A New Method.- Notation.