# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Cartesian currents in the calculus of variations I. Cartesian currents. (English) Zbl 0914.49001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 37. Berlin: Springer. xxiv, 711 p. DM 229.00; öS 1672.00; sFr. 207.00; £88.00; \$ 129.00 (1998).

This book and its companion volume “Cartesian currents in the calculus of variations II. Variational integrals” (see the review below) present a detailed analysis of the so-called Cartesian currents together with their application to specific non-scalar variational problems.

The first volume is divided into five chapters, and to make the book self-contained, Chapter 1 collects the basic facts from general measure theory adressing also topics like weak convergence in ${L}^{1}$ or the theory of Young measures. In Chapter 2 the authors develop the theory of integer multiplicity rectifiable currents. On nearly 100 pages the reader will find all the background material on Geometric Measure Theory starting with the notion of rectifiable sets and ending up with White’s proof of the compactness theorem. After these preparations, Chapter 3 introduces Cartesian maps as vectorfields $u:{\Omega }\to {ℝ}^{N}$ from a domain ${\Omega }$ in ${ℝ}^{n}$ whose Jacobian minors have certain integrability properties and which in addition satisfy $\partial {G}_{u}⌞{\Omega }×{ℝ}^{N}=0$, where ${G}_{u}$ denotes the $n$-current defined as integration of compactly supported differentiable $n$-forms in ${\Omega }×{ℝ}^{N}$ over the graph of $u$ in ${\Omega }$. The authors then prove approximation and compactness theorems for Cartesian maps which require a careful analysis of the weak convergence properties of minors. The last two chapters are devoted to the study of Cartesian currents, i.e., integer multiplicity rectifiable $n$-currents in ${\Omega }×{ℝ}^{N}$ which arise in some sense as limit currrents of smooth graphs. Chapter 4 presents the Euclidean case, in particular, the reader will find closure, compactness and structure theorems. Moreover, a degree theory for Cartesian currents is developed. Chapter 5 deals with the homology theory for currents; the authors discuss topics like Hodge theory, Poincaré-Lefschetz and de Rham dualities and intersection numbers.

The notes at the end of each chapter provide sources of additional information as well as comments on the historical development of the subject. The style of this research monograph is very clear, and a large number of examples together with the selfcontained exposition makes the book readable for any interested graduate student. In summary: “Cartesian currents in the calculus of variations I” can be highly recommended to anybody who wants to be introduced into this new and active field of variational calculus.

##### MSC:
 49-02 Research monographs (calculus of variations) 49Q15 Geometric measure and integration theory, integral and normal currents (optimization) 49Q20 Variational problems in a geometric measure-theoretic setting 26B30 Absolutely continuous functions, functions of bounded variation (several real variables) 58E20 Harmonic maps between infinite-dimensional spaces 74B20 Nonlinear elasticity 76A15 Liquid crystals (fluid mechanics)