## One-dimensional Variational Problems: An IntroductionOne-dimensional variational problems have been somewhat neglected in the literature on calculus of variations, as authors usually treat minimal problems for multiple integrals which lead to partial differential equations and are considerably more difficult to handle. One-dimensional problemsare connected with ordinary differential equations, and hence need many fewer technical prerequisites, but they exhibit the same kind of phenomena and surprises as variational problems for multiple integrals. This book provides an modern introduction to this subject, placing special emphasis ondirect methods. It combines the efforts of a distinguished team of authors who are all renowned mathematicians and expositors. Since there are fewer technical details graduate students who want an overview of the modern approach to variational problems will be able to concentrate on the underlyingtheory and hence gain a good grounding in the subject. Except for results from the theory of measure and integration and from the theory of convex functions, the text develops all mathematical tools needed, including the basic results on one-dimensional Sobolev spaces, absolutely continuousfunctions, and functions of bounded variation. |

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### Contents

Section 1 | 1 |

Section 2 | 7 |

Section 3 | 10 |

Section 4 | 44 |

Section 5 | 46 |

Section 6 | 49 |

Section 7 | 50 |

Section 8 | 51 |

Section 21 | 119 |

Section 22 | 124 |

Section 23 | 126 |

Section 24 | 128 |

Section 25 | 134 |

Section 26 | 149 |

Section 27 | 152 |

Section 28 | 156 |

Section 9 | 54 |

Section 10 | 68 |

Section 11 | 73 |

Section 12 | 82 |

Section 13 | 83 |

Section 14 | 85 |

Section 15 | 86 |

Section 16 | 87 |

Section 17 | 97 |

Section 18 | 104 |

Section 19 | 107 |

Section 20 | 110 |

Section 29 | 166 |

Section 30 | 167 |

Section 31 | 171 |

Section 32 | 179 |

Section 33 | 188 |

Section 34 | 189 |

Section 35 | 203 |

Section 36 | 218 |

Section 37 | 225 |

Section 38 | 236 |

246 | |

### Common terms and phrases

absolutely continuous functions assume assumptions Banach space Borel boundary conditions boundary values bounded variation calculus of variations class C2 classical compact consider converges weakly convex curve deduce defined denotes differential equations direct methods Dirichlet eigenfunction eigenvalue equibounded equivalent Euler equation existence results field on G finite Fp(x functional F graph hence implies inequality integrand interval Lagrangian Lagrangian F(x Lavrentiev phenomenon Lebesgue Lebesgue measure Lemma liminf linear Lipschitz Lipschitz continuous lower semicontinuous Math Mayer field meas measure minimizer of F minimizing sequence minimum problem Moreover necessary condition norm obtain periodic solutions Proof Proposition prove regularity satisfies Section semicontinuous with respect sequentially lower semicontinuous smooth Sobolev spaces Springer subset suitable superlinear growth Suppose Theorem theory Tonelli topology variational integral variational problems weak convergence weak derivative whence zero