Inverse function theorem

Abstract: 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. Holder'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 Kahler Manifolds.- 1. Kahler Manifolds.- 1.1 First Chern Class.- 1.2. Change of Kahler Metrics. Admissible Functions.- 2. Calabi's Conjecture.- 3. Einstein-Kahler Metrics.- 4. Complex Monge-Ampere Equation.- 4.1. About Regularity.- 4.2. About Uniqueness.- 5. Theorem of Existence (the Negative Case).- 6. Existence of Kahler-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-Ampere Equations.- 1. Monge-Ampere 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-Ampere 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.

Abstract: Preface. Chapters: I. Review of fundamental notions of analysis. II. Differential calculus on Banach spaces. III. Differentiable manifolds, finite dimensional case. IV. Integration on manifolds. V. Riemannian manifolds. Kahlerian manifolds. V bis. Connections on a principle fibre bundle. VI. Distributions. VII. Differentiable manifolds, infinite dimensional case. References. Symbols. Index.

Abstract: 1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4.8 The multiplier rule.- 5 Integration.- 5.1 Intervals.- 5.2 Measure.- 5.3 Integrals over En.- 5.4 Integrals over bounded sets.- 5.5 Iterated integrals.- 5.6 Integrals of continuous functions.- 5.7 Change of measure under affine transformations.- 5.8 Transformation of integrals.- 5.9 Coordinate systems in En.- 5.10 Measurable sets and functions further properties.- 5.11 Integrals: general definition, convergence theorems.- 5.12 Differentiation under the integral sign.- 5.13 Lp-spaces.- 6 Curves and line integrals.- 6.1 Derivatives.- 6.2 Curves in En.- 6.3 Differential 1-forms.- 6.4 Line integrals.- *6.5 Gradient method.- *6.6 Integrating factors thermal systems.- 7 Exterior algebra and differential calculus.- 7.1 Covectors and differential forms of degree 2.- 7.2 Alternating multilinear functions.- 7.3 Multicovectors.- 7.4 Differential forms.- 7.5 Multivectors.- 7.6 Induced linear transformations.- 7.7 Transformation law for differential forms.- 7.8 The adjoint and codifferential.- *7.9 Special results for n = 3.- *7.10 Integrating factors (continued).- 8 Integration on manifolds.- 8.1 Regular transformations.- 8.2 Coordinate systems on manifolds.- 8.3 Measure and integration on manifolds.- 8.4 The divergence theorem.- *8.5 Fluid flow.- 8.6 Orientations.- 8.7 Integrals of r-forms.- 8.8 Stokes's formula.- 8.9 Regular transformations on submanifolds.- 8.10 Closed and exact differential forms.- 8.11 Motion of a particle.- 8.12 Motion of several particles.- Axioms for a vector space.- Mean value theorem Taylor's theorem.- Review of Riemann integration.- Monotone functions.- References.- Answers to problems.

Abstract: A global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ ℝn bounded and open, belonging to the Sobolev space W1.p(Ω), p > n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.