## One-dimensional Variational Problems: An IntroductionOne-dimensional variational problems are often neglected in favor of problems which use multiple integrals and partial differential equations, which are typically more difficult to handle. However, these problems and their associated ordinary differential equations do exhibit many of the same challenges and complexity of higher-dimensional problems, while being accessible to more students. This book for graduate students provides the first modern introduction to this subject. It emphasizes direct methods and provides an exceptionally clear view of the underlying theory. Except for standard material on measures, integration and convex functions, the book develops all of the necessary mathematical tools, including basic results for one-dimensional Sobolev spaces, absolutely continuous functions, and functions of bounded variation. |

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### Contents

Section 1 | 10 |

Section 2 | 42 |

Section 3 | 44 |

Section 4 | 46 |

Section 5 | 49 |

Section 6 | 50 |

Section 7 | 51 |

Section 8 | 52 |

Section 12 | 83 |

Section 13 | 104 |

Section 14 | 127 |

Section 15 | 134 |

Section 16 | 156 |

Section 17 | 180 |

Section 18 | 181 |

Section 19 | 195 |

### Common terms and phrases

apply assume assumptions Banach space belongs bounded called choose classical Clearly closed compact conclude consequence consider constant convergence convex curve deduce defined definition denotes derivative differential direct methods discuss eigenvalue equality equivalent Euler equation everywhere example existence extend extremal fact field Finally finite fixed Fp(x given hence holds implies inequality infer instance integral integrand interval introduce Lagrangian Lebesgue Lemma lim inf linear Lipschitz Lipschitz-continuous lower semicontinuous means meas measure minimizer minimum problem Moreover necessary Note obtain optimal particular periodic positive Proof Proposition prove question regularity Remark respect result satisfies sequence sequentially smooth Sobolev spaces solution solves string subsequence sufficient suitable Suppose taking Theorem theory Tonelli's true uniformly unique values variational integral variational problems weak weakly whence yields zero