Cartesian currents in the calculus of variations II. Variational integrals.

*(English)*Zbl 0914.49002
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 38. Berlin: Springer. xxiv, 697 p. (1998).

The second volume of the monograph “Cartesian currents in the calculus of variations” is mainly devoted to the study of specific non-scalar variational problems with the help of the tools which have been developed in volume I (see the review above). The book is divided into six chapters, and these chapters are readable independently of each other.

The first chapter discusses the general setting, i.e., the notion of regular variational integrals in the class of Cartesian currents. In Chapter 2 the authors introduce the basic concepts of nonlinear elasticity, for example, they explain in great detail different forms of stored energies and discuss constitutive conditions. After these preparations the weak diffeomorphisms are shown to serve as the correct mathematical model for elastic deformations. At the end of Chapter 2 it is shown that various boundary value problems from nonlinear elasticity are solvable within subclasses of weak diffeomorphisms. Chapter 3 is devoted to the Dirichlet integral in Sobolev spaces, which means that, roughly speaking, the authors collect “all” known existence and regularity results for weakly harmonic maps between Riemannian manifolds. For example, the reader will find the partial regularity result of Schoen-Uhlenbeck for enery minimizing harmonic maps as well as results on energy minimizing maps with prescribed action or loops. As the authors write in the introduction of Chapter 4 they here begin to develop a natural approach to variational problems for the Dirichlet energy in terms of Cartesian currents. To be specific, the Dirichlet energy for maps into \(S^2\) is studied under the assumption that the domain of definition is a region in \(\mathbb{R}^2\) or \(\mathbb{R}^3\). In this setting the Dirichlet integral is first extended to Cartesian currents, then the existence of minimizing currents is established and finally partial regularity of minimizers is shown. Chapter 5 generalizes some of the methods and results of the previous chapter to more general situations, for example, the liquid crystal energy is studied first in the classical way via the Sobolev space approach, then the relaxed version is minimized on a suitable set of Cartesian currents. The final Chapter 6 adresses the non-parametric area functional: starting from the classical theory of codimension one surfaces of least area (in various settings), the authors then discuss \(n\)-dimensional graphs in \(\mathbb{R}^n\times \mathbb{R}^N\) in the framework of Cartesian currents.

As in the first volume each chapter of volume II is followed by a collection of remarks adressing related problems and further results and also giving historical comments.

Although the subject is very deep, the interested reader can easily follow the clear lines of exposition, and a large number of examples helps to illustrate the theory. A detailed table of contents, an extensive index and a list of 686 references also support the reader in consulting this monograph. In summary: “Cartesian currents in the calculus of variations II. Variational integrals” can be highly recommended to anybody including graduate students and active researchers who is interested in this new and active field of variational calculus.

The first chapter discusses the general setting, i.e., the notion of regular variational integrals in the class of Cartesian currents. In Chapter 2 the authors introduce the basic concepts of nonlinear elasticity, for example, they explain in great detail different forms of stored energies and discuss constitutive conditions. After these preparations the weak diffeomorphisms are shown to serve as the correct mathematical model for elastic deformations. At the end of Chapter 2 it is shown that various boundary value problems from nonlinear elasticity are solvable within subclasses of weak diffeomorphisms. Chapter 3 is devoted to the Dirichlet integral in Sobolev spaces, which means that, roughly speaking, the authors collect “all” known existence and regularity results for weakly harmonic maps between Riemannian manifolds. For example, the reader will find the partial regularity result of Schoen-Uhlenbeck for enery minimizing harmonic maps as well as results on energy minimizing maps with prescribed action or loops. As the authors write in the introduction of Chapter 4 they here begin to develop a natural approach to variational problems for the Dirichlet energy in terms of Cartesian currents. To be specific, the Dirichlet energy for maps into \(S^2\) is studied under the assumption that the domain of definition is a region in \(\mathbb{R}^2\) or \(\mathbb{R}^3\). In this setting the Dirichlet integral is first extended to Cartesian currents, then the existence of minimizing currents is established and finally partial regularity of minimizers is shown. Chapter 5 generalizes some of the methods and results of the previous chapter to more general situations, for example, the liquid crystal energy is studied first in the classical way via the Sobolev space approach, then the relaxed version is minimized on a suitable set of Cartesian currents. The final Chapter 6 adresses the non-parametric area functional: starting from the classical theory of codimension one surfaces of least area (in various settings), the authors then discuss \(n\)-dimensional graphs in \(\mathbb{R}^n\times \mathbb{R}^N\) in the framework of Cartesian currents.

As in the first volume each chapter of volume II is followed by a collection of remarks adressing related problems and further results and also giving historical comments.

Although the subject is very deep, the interested reader can easily follow the clear lines of exposition, and a large number of examples helps to illustrate the theory. A detailed table of contents, an extensive index and a list of 686 references also support the reader in consulting this monograph. In summary: “Cartesian currents in the calculus of variations II. Variational integrals” can be highly recommended to anybody including graduate students and active researchers who is interested in this new and active field of variational calculus.

Reviewer: M.Fuchs (Saarbrücken)

##### MSC:

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

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

49Q20 | Variational problems in a geometric measure-theoretic setting |

26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |

58E20 | Harmonic maps, etc. |

74B20 | Nonlinear elasticity |

76A15 | Liquid crystals |