Author | Aubin, Thierry. author |
---|---|

Title | Nonlinear Analysis on Manifolds. Monge-Ampรจre Equations [electronic resource] / by Thierry Aubin |

Imprint | New York, NY : Springer New York, 1982 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-5734-9 |

Descript | XII, 204 p. online resource |

SUMMARY

This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis

CONTENT

1 Riemannian Geometry -- ยง1. Introduction to Differential Geometry -- ยง2. Riemannian Manifold -- ยง3. Exponential Mapping -- ยง4. The Hopf-Rinow Theorem -- ยง5. Second Variation of the Length Integral -- ยง6. Jacobi Field -- ยง7. The Index Inequality -- ยง8. Estimates on the Components of the Metric Tensor -- ยง9. Integration over Riemannian Manifolds -- ยง10. Manifold with Boundary -- ยง11. Harmonic Forms -- 2 Sobolev Spaces -- ยง1. First Definitions -- ยง2. Density Problems -- ยง3. Sobolev Imbedding Theorem -- ยง4. Sobolevโ{128}{153}s Proof -- ยง5. Proof by Gagliardo and Nirenberg -- ยง6. New Proof -- ยง7. Sobolev Imbedding Theorem for Riemannian Manifolds -- ยง8. Optimal Inequalities -- ยง9. Sobolevโ{128}{153}s Theorem for Compact Riemannian Manifolds with Boundary -- ยง10. The Kondrakov Theorem -- ยง11. Kondrakovโ{128}{153}s Theorem for Riemannian Manifolds -- ยง12. Examples -- ยง13. Improvement of the Best Constants -- ยง14. The Case of the Sphere -- ยง15. The Exceptional Case of the Sobolev Imbedding Theorem -- ยง16. Moserโ{128}{153}s Results -- ยง17. The Case of the Riemannian Manifolds -- ยง18. Problems of Traces -- 3 Background Material -- ยง1. Differential Calculus -- ยง2. Four Basic Theorems of Functional Analysis -- ยง3. Weak Convergence. Compact Operators -- ยง4. The Lebesgue Integral -- ยง5. The LpSpaces -- ยง6. Elliptic Differential Operators -- ยง7. Inequalities -- ยง8. Maximum Principle -- ยง9. Best Constants -- 4 Greenโ{128}{153}s Function for Riemannian Manifolds -- ยง1. Linear Elliptic Equations -- ยง2. Greenโ{128}{153}s Function of the Laplacian -- 5 The Methods -- ยง1. Yamabeโ{128}{153}s Equation -- ยง2. Bergerโ{128}{153}s Problem -- ยง3. Nirenbergโ{128}{153}s Problem -- 6 The Scalar Curvature -- ยง1. The Yamabe Problem -- ยง2. The Positive Case -- ยง3. Other Problems -- 7 Complex Monge-Ampere Equation on Compact Kรคhler Manifolds -- ยง1. Kรคhler Manifolds -- ยง2. Calabiโ{128}{153}s Conjecture -- ยง3. Einstein-Kรคhler Metrics -- ยง4. Complex Monge-Ampere Equation -- ยง5. Theorem of Existence (the Negative Case) -- ยง6. Existence of Kรคhler-Einstein Metric -- ยง7. Theorem of Existence (the Null Case) -- ยง8. Proof of Calabiโ{128}{153}s Conjecture -- ยง9. The Positive Case -- ยง10. A Priori Estimate for ?? -- ยง11. A Priori Estimate for the Third Derivatives of Mixed Type -- ยง12. The Method of Lower and Upper Solutions -- 8 Monge-Ampรจre Equations -- ยง1. Monge-Ampรจre Equations on Bounded Domains of ?n -- ยง2. The Estimates -- ยง3. The Radon Measure ?(?) -- ยง4. The Functional ? (?) -- ยง5. Variational Problem -- ยง6. The Complex Monge-Ampรจre Equation -- ยง7. The Case of Radially Symmetric Functions -- ยง8. A New Method -- Notation

Mathematics
Matrix theory
Algebra
Mathematics
Linear and Multilinear Algebras Matrix Theory