## Relaxation in Optimization Theory and Variational CalculusIntroduces applied mathematicians and graduate students to an original relaxation method based on a continuous extension of various optimization problems relating to convex compactification; it can be applied to problems in optimal control theory, the calculus of variations, and non-cooperative game theory. Reviews the background and summarizes the general theory of convex compactifications, then uses it to obtain convex, locally compact envelopes of the Lebesague and Sobolev spaces involved in concrete problems. The nontrivial envelopes cover the classical Young measures as well as various generalizations of them, which can record the limit behavior of fast oscillation and concentration effects. Annotation copyrighted by Book News, Inc., Portland, OR |

### Contents

1 | 1 |

2 | 36 |

Theory of convex compactifications | 81 |

Young measures and their generalizations | 102 |

Convex compactifications of balls in LPspaces | 151 |

Convex σcompactifications of LPspaces | 173 |

Approximation theory | 188 |

Extensions of Nemytskii mappings | 202 |

96 | 312 |

Relaxation in variational calculus I | 320 |

3 | 348 |

4 | 366 |

Relaxation scheme and its FEMapproximation | 383 |

an inner case | 391 |

Relaxation in game theory | 412 |

120 | 445 |

### Common terms and phrases

a₁ adjoint admissible approximation B-coercive Banach space bounded boundedness cluster point coercivity compact constraints continuous extension convergence convex compactification cost functional data qualification defined denotes differential DiPerna-Majda measures equation estimate Example finer Gâteaux differentiable H₁ H₂ Hamiltonian homeomorphism homogeneous imbedding K₁ L¹(N L²(N Lª(N Lebesgue Lebesgue spaces Lemma Limsup linear subspace locally convex space lower semicontinuous LP Q maximum principle Minimize Moreover Nash equilibrium Nemytskii mapping nonlinear norm Note o-compactification optimal control optimal control problems optimality conditions p-nonconcentrating Proof Proposition Pvvc quasiconvex rank-one relaxed problem respectively RH Pvc Rmxn RPvc RPvvc S₁ satisfied Section sequence solution subset suppose Theorem topological space topology u₁ U₂ V₁ valid variational problems W¹P weak weak topology weakly Young functionals Young measure