In recent years there has been a considerable renewal of interest in the clas­ sical problems of the calculus of variations, both from the point of view of mathematics and of applications. Some of the most powerful tools for proving existence of minima for such problems are known as direct methods. They are often the only available ones, particularly for vectorial problems. It is the aim of this book to present them. These methods were introduced by Tonelli, following earlier work of Hilbert and Lebesgue. Although there are excellent books on calculus of variations and on direct methods, there are recent important developments which cannot be found in these books; in particular, those dealing with vector valued functions and relaxation of non convex problems. These two last ones are important in appli­ cations to nonlinear elasticity, optimal design . . . . In these fields the variational methods are particularly effective. Part of the mathematical developments and of the renewal of interest in these methods finds its motivations in nonlinear elasticity. Moreover, one of the recent important contributions to nonlinear analysis has been the study of the behaviour of nonlinear functionals un­ der various types of convergence, particularly the weak convergence. Two well studied theories have now been developed, namely f-convergence and compen­ sated compactness. They both include as a particular case the direct methods of the calculus of variations, but they are also, both, inspired and have as main examples these direct methods.

1 Introduction.- 1.1 General Considerations and Some Examples.- 1.1.1 Statement of the Problem and Some Examples.- 1.1.2 The Classical Approach.- 1.1.3 Direct Methods.- 1.2 The Direct Methods.- 1.2.1 Preliminaries.- 1.2.2 Abstract Results and the Scalar Case.- 1.2.3 The Vectorial Case.- 1.2.4 Nonconvex Integrands.- 1.2.5 Applications to Nonlinear Elasticity.- 2 Preliminaries.- 2.1 Lp and Sobolev Spaces.- 2.1.1 Weak Convergence in Lp.- 2.1.2 Sobolev Spaces.- 2.2 Convex Analysis.- 2.2.1 Convex Functions.- 2.2.2 Duality and Hahn-Banach Theorem.- 2.2.3 Carathéodory Theorem.- 3 General Setting and the Scalar Case.- 3.0 Introduction.- 3.1 Abstract Results.- 3.1.1 Weak Lower Semicontinuous Functionals and Existence Theorems.- 3.1.2 Convex Functionals.- 3.1.3 First-Order Necessary Condition.- 3.2 Convex Functionals.- 3.2.1 Necessary Condition.- 3.2.2 Sufficient Condition.- 3.3 Weak Lower Semicontinuity, Weak Continuity and Invariant Integrals.- 3.3.1 Weak Lower Semicontinuity.- 3.3.2 Weak Continuity and Invariant Integrals.- 3.4 Existence Theorems and Euler Equations.- 3.4.1 Existence Theorems and Regularity Results.- 3.4.2 Euler Equations.- 3.4.3 Lavrentiev Phenomenon.- 4 The Vectorial Case.- 4.0 Introduction.- 4.1 Polyconvexity, Quasi convexity and Rank One Convexity.- 4.1.1 Definitions and Properties.- 4.1.2 Examples.- 4.2 Weak Continuity, Weak Lower Semicontinuity and Existence Theorems.- 4.2.1 Weak Lower Semicontinuity.- 4.2.2 Weak Continuity.- 4.2.3 Existence Theorems.- 4.3 Appendix: Some Elementary Properties of Determinants.- 4.3.1 Definitions and Properties of Determinants.- 4.3.2 Some Properties of Jacobians.- 5 Non-Convex Integrands.- 5.0 Introduction.- 5.1 Convex, Poly con vex, Quasiconvex, Rank One Convex Envelopes.- 5.1.1 Definitions and Properties.- 5.1.2 Examples.- 5.2 Relaxation Theorems.- 5.2.1 Relaxation Theorems.- 5.2.2 Existence and Non-Existence of Solutions.- Appendix: Applications.- A.0 Introduction.- A.1 Existence and Uniqueness Theorems in Nonlinear Elasticity.- A.1.1 Setting of the Problem.- A.1.2 Existence Theorems.- A.1.3 Unicity of Classical Solutions of Equilibrium Equations.- A.2 Relaxation Theorems in Elasticity and Optimal Design.- A.2.1 Antiplane Shear Problem in Elasticity.- A.2.2 A Problem of Equilibrium of Gases.- A.2.3 Equilibrium of Elastic Crystals.- A.2.4 Relaxation and Optimal Design.