Comparison principles for general potential theories and PDEs
Cover -- Contents -- Preface -- Guide for the Reader -- I. A Comprehensive Introduction -- 1. A Comprehensive Introduction -- 1.1 The Potential Theory Setting -- 1.2 The Differential Operator Setting -- 1.3 The Correspondence Principle -- 1.4 Canonical Operators -- 1.5 Gradient-Free Operators -- 1.6...
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| Hauptverfasser: | , , , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton ; Oxford
Princeton University Press
[2023]
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| Schriftenreihe: | Annals of Mathematics Studies
Number 218 |
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| Zusammenfassung: | Cover -- Contents -- Preface -- Guide for the Reader -- I. A Comprehensive Introduction -- 1. A Comprehensive Introduction -- 1.1 The Potential Theory Setting -- 1.2 The Differential Operator Setting -- 1.3 The Correspondence Principle -- 1.4 Canonical Operators -- 1.5 Gradient-Free Operators -- 1.6 Operators Involving Gårding-Dirichlet Polynomials -- 1.7 General Potential-Theoretic Comparison Theorems -- 1.8 Limitations of the Method and Comparison with the Literature -- 1.9 Reflections on "Potential Theory versus Operator Theory" -- II. The Potential Theory Approach -- 2. Constant-Coefficient Constraint Sets and Their Subharmonics -- 3. Dirichlet Duality and F-Subharmonic Functions -- 4. Monotonicity Cones for Constant-Coefficient Subequations -- 4.1 The Maximal Monotonicity Cone Subequation -- 4.2 Product Monotonicity Cone Subequations -- 5. A Fundamental Family of Monotonicity Cone Subequations -- 5.1 Construction of the Fundamental Family -- 5.2 Nesting, Limit Cases, and Simplifying the Family of Cones -- 5.3 The Fundamental Nature of the Family of Monotonicity Cones -- 6. The Zero Maximum Principle for Dual Monotonicity Cones -- 7. Comparison Principles for Potential Theories with Sufficient Monotonicity -- 8. Comparison on Arbitrary Domains by Additional Monotonicity -- 9. Failure of Comparison with Insufficient Maximal Monotonicity -- 9.1 Finite R and Failure of Comparison on Large Domains -- 9.2 Failure of Comparison on Arbitrarily Small Domains -- 10. Special Cases: Reduced Constraint Sets -- 10.1 Pure Second Order -- 10.2 Gradient-Free -- 10.3 First Order and Pure First Order -- 10.4 Zero-Order-Free -- 10.5 Summary -- III. Marrying Potential Theory to Operator Theory via the Correspondence Principle -- 11. The Correspondence Principle for Compatible Operator-Subequation Pairs. |
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| Beschreibung: | xiv, 203 Seiten |
| ISBN: | 9780691243610 9780691243627 |