## Partial Differential EquationsThis book is a very well-accepted introduction to the subject. In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis. Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation without solutions. |

### Contents

Chapter | 1 |

Quasilinear Equations | 9 |

Chapter 3 | 39 |

Chapter 8 | 53 |

Characteristic Manifolds and the Cauchy Problem | 54 |

The Cauchy Problem | 58 |

The LagrangeGreen Identity | 79 |

Distribution Solutions | 89 |

45 | 137 |

HigherOrder Hyperbolic Equations with Constant Coefficients | 143 |

www | 150 |

Symmetric Hyperbolic Systems | 163 |

37 | 175 |

Chapter 6 | 185 |

More on the Hilbert Space H and the Assumption of Boundary Values | 198 |

Chapter 7 | 206 |

The Maximum Principle | 103 |

Proof of Existence of Solutions for the Dirichlet Problem Using | 111 |

Solution of the Dirichlet Problem by HilbertSpace Methods | 117 |

Chapter 5 | 126 |

The OneDimensional Wave Equation | 128 |

The InitialValue Problem for General SecondOrder Linear | 227 |

H Lewys Example of a Linear Equation | 235 |

Bibliography | 241 |

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### Common terms and phrases

assume ball belongs bounded Cauchy data Cauchy problem Cauchy sequence Chapter characteristic curves class Cē coefficients compact subset compact support complex cone constant continuous converge defined denote derivatives of orders determined uniquely Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f fundamental solution given harmonic function hence Hilbert space Hint holds identity implies inequality initial conditions initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix maximum principle neighborhood non-characteristic obtained open set ordinary differential equations partial differential equation plane polynomial power series prescribed proof quasi-linear real analytic functions real numbers satisfies scalar second derivatives Show solved sufficiently small test function u₁ uniformly vanish vector wave equation x₁ ΘΩ ди