Introduction Chapter XVII. Second Order Elliptic Operators Summary 17.1. Interior Regularity and Local Existence Theorems 17.2. Unique Continuation Theorems 17.3. The Dirichlet Problem 17.4. The Hadamard Parametrix Construction 17.5. Asymptotic Properties of Eigenvalues and Eigenfunctions Notes Chapter XVIII. Pseudo-Differential Operators Summary 18.1. The Basic Calculus 18.2. Conormal Distributions 18.3. Totally Characteristic Operators 18.4. Gauss Transforms Revisited 18.5. The Weyl Calculus 18.6. Estimates of Pseudo-Differential Operators Notes Chapter XIX. Elliptic Operators on a Compact Manifold Without Boundary Summary 19.1. Abstract Fredholm Theory 19.2. The Index of Elliptic Operators 19.3. The Index Theorem in 19.4. The Lefschetz Formula 19.5. Miscellaneous Remarks on Ellipticity Notes Chapter XX. Boundary Problems for Elliptic Differential Operators . Summary 20.1. Elliptic Boundary Problems 20.2. Preliminaries on Ordinary Differential Operators 20.3. The Index for Elliptic Boundary Problems 20.4. Non-Elliptic Boundary Problems Notes Chapter XXI. Symplectic Geometry Summary 21.1. The Basic Structure 21.2. Submanifolds of a Sympletic Manifold 21.3. Normal Forms of Functions 21.4. Folds and Glancing Hypersurfaces 21.5. Symplectic Equivalence of Quadratic Forms 21.6. The Lagrangian Grassmannian Notes Chapter XXII. Some Classes of (Micro-)hypoelliptic Operators Summary 22.1. Operators with Pseudo-Differential Parametrix 22.2. Generalized Kolmogorov Equations 22.3. Melin's Inequality 22.4. Hypoellipticity with Loss of One Derivative Notes Chapter XXIII. The Strictly hyperbolic Cauchy Problem
Summary 23.1. First Order Operators 23.2. Operators of Higher Order 23.3. Necessary Conditions for Correctness of the Cauchy Problem 23.4. Hyperbolic Operators of Principal Type Notes Chapter XXIV. The Mixed Dirichlet-Cauchy Problem for Second Order Operators Summary 24.1. Energy Estimates and Existence Theorems in the Hyperbolic Case 24.2. Singularities in the Elliptic and Hyperbolic Regions 24.3. The Generalized Bicharacteristic Flow 24.4. The Diffractive Case 24.5. The General Propagation of Singularities 24.6. Operators Microlocally of Tricomi's Type 24.7. Operators Depending on Parameters Notes Appendix B. Some Spaces of Distributions B.1 Distributions in 1Rn and in an Open Manifold B.2. Distributions in a Half Space and in a Manifold with Boundary Appendix C. Some Tools from Differential Geometry C.1. The Frobenius Theorem and Foliations C.2. A Singular Differential Equation C.3. Clean Intersections and Maps of Constant Rank C.4. Folds and Involutions C.5. Geodesic Normal Coordinates C.6. The Morse Lemma with Parameters Notes Bibliography Index Index of Notation
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