## Direct Methods in the Calculus of VariationsThis book is developed for the study of vectorial problems in the calculus of variations. The subject is a very active one and almost half of the book consists of new material. This is a new edition of the earlier book published in 1989 and it is suitable for graduate students. The book has been updated with some new material and examples added. Applications are included. |

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### Contents

7 | |

20 | |

Convex analysis and the scalar case | 29 |

Lower semicontinuity and existence theorems | 73 |

The one dimensional case 119 | 118 |

5 | 153 |

Polyconvex quasiconvex and rank one convex envelopes | 265 |

Lower semi continuity and existence theorems in | 367 |

Existence of minima for nonquasiconvex integrands | 465 |

Function spaces | 503 |

Singular values | 514 |

Some underdetermined partial differential equations | 534 |

Extension of Lipschitz functions on Banach spaces | 548 |

Bibliography | 569 |

Notation | 576 |

592 | |

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### Common terms and phrases

afﬁne assume bounded open set calculus of variations Carathéodory function Chapter choose claim compact convex functions convex hull convex set convexity of f Corollary Dacorogna Dacorogna-Marcellini deduce deﬁne deﬁnition divide the proof E C RN equivalent Euler-Lagrange equation example exists u G f is convex fact ﬁnd ﬁnite ﬁrst function f G int G R2 G RN G RNX hence holds hypothesis implies inequality integrals Jensen inequality Lemma Let f Let Q C R Lipschitz boundary locally bounded lower semicontinuous Math matrix meas Q measQ minimizer n)-invariant notation Observe obtain PcoE polyconvex problem property for contractions Proposition prove Q;RN quasiafﬁne quasiconvex functions rank one convex RcoE recall Remark result RN×n Rnxn satisﬁes satisfying Section separately convex singular values Sobolev spaces solution Step strictly convex Theorem 5.6 vectorial