Direct methods in the calculus of variations.
2nd ed.

*(English)*Zbl 1140.49001
Applied Mathematical Sciences 78. Berlin: Springer (ISBN 978-0-387-35779-9/hbk). xii, 619 p. (2008).

The present monograph has been announced by the author as a “revised and augmented edition to Direct Methods in the Calculus of Variations”. It is centered around lower semicontinuity and relaxation theorems for problems in multidimensional calculus of variations in their interplay with generalized convexity notions. The contents of the first edition are complemented with all essential discoveries and developments in this very active area since 1989. In spite oft the wealth of material and the fair difficulty of the subject, the author maintains a fresh and lucid style, resulting in a concise, very well readable presentation. Surely this book will define a long-lasting standard in its area.

The monograph starts with an introductory chapter, which is a short review of the book itself. Here the basic concepts are introduced, and even the main theorems are stated (Chapter 1). In multidimensional calculus of variations, the fundamental distinction is between the “scalar case”, comprising variational problems for unknown functions \(u(x) :\mathbb R^n \to \mathbb R^N\) with \(N=1\) or \(n =1\), and the “vectorial case” where \(N \geq 2\) and \(n \geq 2\). While in the scalar case existence and relaxation theorems are connected with the usual convexity, the vectorial case requires the study of generalized convexity notions as polyconvexity, quasiconvexity (in Morrey’s sense) and rank one convexity.

In the framework of the scalar case, the author starts with a summary of convex analysis (Chapter 2). Then weak lower semicontinuity theorems, invariant integrals, existence theorems and the Euler-Lagrange equations in the weak and classical form are presented (Chapter 3). The following chapter treats in some brevity the special case \(n=1\): existence, necessary conditions, Hamiltonian formulation, regularity and the Lavrentiev phenomenon (Chapter 4).

In the vectorial case, generalized convex functions (Chapter 5) and generalized convex envelopes (Chapter 6) are studied. An exhaustive list of examples is presented and discussed. The following application of generalized convexity notions to subsets of \(\mathbb R^{N \times n}\), introducing “quasiconvex analysis” as a generalization of the geometrical concepts of convex analysis (Chapter 7), is surely among the highlights of the book. Then the lower semicontinuity and existence theorems for the vectorial case are presented (Chapter 8).

A third part of the book is concerned with the treatment of nonconvex (resp. non-quasiconvex) variational problems. First, the author proves relaxation theorems (Chapter 9) whereby Young measure techniques have been avoided. Then existence theorems for implicit first-order PDE’s (Chapter 10) and existence theorems for diverse variational problems with non-quasiconvex integrands (Chapter 11) are presented.

The book is completed by a series of chapters entitled “Miscellaneous”. The topics collected here attract interest on its own and are, as a rule, as yet underrepresented in the literature. A chapter entitled “Function spaces” covers Sobolev as well as Hölder spaces (Chapter 12). Since a lot of examples is concerned with singular values instead of eigenvalues, a special chapter is devoted to them (Chapter 13). Then the theory of Dirichlet BVP’s for the first-order PDE’s \(\text{div }u = f\), \(\text{curl } u= f\) and \(\text{det } (\nabla u) =f\) is summarized (Chapter 14). Finally, the author’s results on the extension of Lipschitz functions defined on Banach spaces are presented (Chapter 15).

The exhaustive bibliography comprises 621 references and covers the relevant publications in the area until 2007.

The monograph starts with an introductory chapter, which is a short review of the book itself. Here the basic concepts are introduced, and even the main theorems are stated (Chapter 1). In multidimensional calculus of variations, the fundamental distinction is between the “scalar case”, comprising variational problems for unknown functions \(u(x) :\mathbb R^n \to \mathbb R^N\) with \(N=1\) or \(n =1\), and the “vectorial case” where \(N \geq 2\) and \(n \geq 2\). While in the scalar case existence and relaxation theorems are connected with the usual convexity, the vectorial case requires the study of generalized convexity notions as polyconvexity, quasiconvexity (in Morrey’s sense) and rank one convexity.

In the framework of the scalar case, the author starts with a summary of convex analysis (Chapter 2). Then weak lower semicontinuity theorems, invariant integrals, existence theorems and the Euler-Lagrange equations in the weak and classical form are presented (Chapter 3). The following chapter treats in some brevity the special case \(n=1\): existence, necessary conditions, Hamiltonian formulation, regularity and the Lavrentiev phenomenon (Chapter 4).

In the vectorial case, generalized convex functions (Chapter 5) and generalized convex envelopes (Chapter 6) are studied. An exhaustive list of examples is presented and discussed. The following application of generalized convexity notions to subsets of \(\mathbb R^{N \times n}\), introducing “quasiconvex analysis” as a generalization of the geometrical concepts of convex analysis (Chapter 7), is surely among the highlights of the book. Then the lower semicontinuity and existence theorems for the vectorial case are presented (Chapter 8).

A third part of the book is concerned with the treatment of nonconvex (resp. non-quasiconvex) variational problems. First, the author proves relaxation theorems (Chapter 9) whereby Young measure techniques have been avoided. Then existence theorems for implicit first-order PDE’s (Chapter 10) and existence theorems for diverse variational problems with non-quasiconvex integrands (Chapter 11) are presented.

The book is completed by a series of chapters entitled “Miscellaneous”. The topics collected here attract interest on its own and are, as a rule, as yet underrepresented in the literature. A chapter entitled “Function spaces” covers Sobolev as well as Hölder spaces (Chapter 12). Since a lot of examples is concerned with singular values instead of eigenvalues, a special chapter is devoted to them (Chapter 13). Then the theory of Dirichlet BVP’s for the first-order PDE’s \(\text{div }u = f\), \(\text{curl } u= f\) and \(\text{det } (\nabla u) =f\) is summarized (Chapter 14). Finally, the author’s results on the extension of Lipschitz functions defined on Banach spaces are presented (Chapter 15).

The exhaustive bibliography comprises 621 references and covers the relevant publications in the area until 2007.

Reviewer: Marcus Wagner (Cottbus)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

26B25 | Convexity of real functions of several variables, generalizations |

26B35 | Special properties of functions of several variables, Hölder conditions, etc. |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

49J10 | Existence theories for free problems in two or more independent variables |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49K10 | Optimality conditions for free problems in two or more independent variables |