Direct methods in the calculus of variations.

*(English)*Zbl 0703.49001
Applied Mathematical Sciences, 78. Berlin etc.: Springer-Verlag. ix, 308 p. DM 120.00/hbk (1989).

From the author’s preface: “In recent years there has been a considerable renewal of interest in the classical 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 applications to nonlinear elasticity, optimal design.... In these fields the variational methods are particularly effective. Parts of the mathematical developments and of the renewal of interest in these methods find its motivations in nonlinear elasticity. Moreover, one of the recent important contributions to nonlinear analysis has been the study of the behavior of nonlinear functionals under various types of convergence, particularly weak convergence. Two well studied theories have now been developed, namely \(\Gamma\)-convergence and compensated 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.”

A brief description of the contents of the monograph is given in the first introductory chapter. The second chapter is devoted to the necessary background for the succeeding four parts: weak convergence in \(L^ p\), Sobolev spaces and convex analysis. The necessary facts are listed and some of them are proved.

The general and fundamental results concerning with direct methods are described in the next three chapters. The investigation is done for the scalar case and for the vectorial case. These cases have substantially different features. Essential contributions to the development of the discussed theory are due to the author.

The third chapter is devoted to the scalar case. The author shows that the minimization problem in a reflexive Banach space has at least one solution if the functional is weakly lower semicontinuous and coercive. It is proved that convex functionals are weakly lower semicontinuous. The classical first order necessary minimality condition is presented under the assumption that the functional has a Gateaux derivative. These facts are used for an investigation of minimization problems in the calculus of variations. Necessary and sufficient conditions ensuring convexity, weakly lower semicontinuity, Gateaux differentiability of functionals in Sobolev spaces are derived. These results make up the foundations for finding the existence conditions and for establishing and investigating the Euler equations.

The fourth chapter is devoted to the vectorial case. As it is shown in the previous chapter the convexity of the integrand with respect to derivatives is playing the central role in the scalar case. It turns out that in the vectorial case this condition is still sufficient for weak lower semicontinuity of the integral functional in a Sobolev space, but it is far from being a necessary condition. In order to obtain similar necessary and sufficient condition in the vectorial case the author considers polyconvex, quasiconvex and rank one convex functions. These notions and their relationships are investigated and existence theorems are given. It is shown that quasiconvexity plays a central role.

In the fifth chapter the author investigates variational problems in which the integrand fails to be convex in the scalar case and quasiconvex in the vectorial case. In general, such problems have no solution in a Sobolev space. In order to introduce the so-called relaxed problem the author derives the notion of convex, polyconvex, quasiconvex and rank one convex envelopes. Then he introduces the relaxed problem in which the integrand is the quasiconvex envelope of the integrand in the initial problem. Properties of the envelopes are investigated and relaxation results are presented.

Finally, in the appendix, applications of the obtained results to nonlinear elasticity and optimal design are given.

Many examples and counterexamples accompany the text to show main properties of derived notions and to illustrate the obtained results. Thus, the book summarizes and unifies the well-known classical results in the theory of direct methods and the important investigations made up to today. Undoubtedly, it will win recognition of specialists and will get an important place in the scientific literature. It is a useful foundation for study, research and applications.

A brief description of the contents of the monograph is given in the first introductory chapter. The second chapter is devoted to the necessary background for the succeeding four parts: weak convergence in \(L^ p\), Sobolev spaces and convex analysis. The necessary facts are listed and some of them are proved.

The general and fundamental results concerning with direct methods are described in the next three chapters. The investigation is done for the scalar case and for the vectorial case. These cases have substantially different features. Essential contributions to the development of the discussed theory are due to the author.

The third chapter is devoted to the scalar case. The author shows that the minimization problem in a reflexive Banach space has at least one solution if the functional is weakly lower semicontinuous and coercive. It is proved that convex functionals are weakly lower semicontinuous. The classical first order necessary minimality condition is presented under the assumption that the functional has a Gateaux derivative. These facts are used for an investigation of minimization problems in the calculus of variations. Necessary and sufficient conditions ensuring convexity, weakly lower semicontinuity, Gateaux differentiability of functionals in Sobolev spaces are derived. These results make up the foundations for finding the existence conditions and for establishing and investigating the Euler equations.

The fourth chapter is devoted to the vectorial case. As it is shown in the previous chapter the convexity of the integrand with respect to derivatives is playing the central role in the scalar case. It turns out that in the vectorial case this condition is still sufficient for weak lower semicontinuity of the integral functional in a Sobolev space, but it is far from being a necessary condition. In order to obtain similar necessary and sufficient condition in the vectorial case the author considers polyconvex, quasiconvex and rank one convex functions. These notions and their relationships are investigated and existence theorems are given. It is shown that quasiconvexity plays a central role.

In the fifth chapter the author investigates variational problems in which the integrand fails to be convex in the scalar case and quasiconvex in the vectorial case. In general, such problems have no solution in a Sobolev space. In order to introduce the so-called relaxed problem the author derives the notion of convex, polyconvex, quasiconvex and rank one convex envelopes. Then he introduces the relaxed problem in which the integrand is the quasiconvex envelope of the integrand in the initial problem. Properties of the envelopes are investigated and relaxation results are presented.

Finally, in the appendix, applications of the obtained results to nonlinear elasticity and optimal design are given.

Many examples and counterexamples accompany the text to show main properties of derived notions and to illustrate the obtained results. Thus, the book summarizes and unifies the well-known classical results in the theory of direct methods and the important investigations made up to today. Undoubtedly, it will win recognition of specialists and will get an important place in the scientific literature. It is a useful foundation for study, research and applications.

Reviewer: E.G.Al’brekht

##### MSC:

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

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

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