Preface |
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Introduction: What are Partial Differential Equations? |
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1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order |
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2 The Maximum Principle |
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3 Existence Techniques I: Methods Based on the Maximum Principle |
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4 Existence Techniques II: Parabolic Methods. The Heat Equation |
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5 Reaction-Diffusion Equations and Systems |
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6 Hyperbolic Equations |
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7 The Heat Equation, Semigroups, and Brownian Motion.- 8 Relationships between Different Partial Differential Equations |
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9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III) |
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10 Sobolev Spaces and L^2 Regularity theory |
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11 Strong solutions |
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12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV) |
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13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash |
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Appendix: Banach and Hilbert spaces. The L^p-Spaces |
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References |
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Index of Notation |
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Index |
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Preface |
|
Introduction: What are Partial Differential Equations? |
|
1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order |
|
2 The Maximum Principle |
|
3 Existence Techniques I: Methods Based on the Maximum Principle |
|
4 Existence Techniques II: Parabolic Methods. The Heat Equation |
|