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 $u:\Omega\to \bbfR^N$ from a domain $\Omega$ in $\bbfR^n$ whose Jacobian minors have certain integrability properties and which in addition satisfy $\partial G_u \llcorner \Omega\times \bbfR^N= 0$, where $G_u$ denotes the $n$-current defined as integration of compactly supported differentiable $n$-forms in $\Omega\times \bbfR^N$ 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 $\Omega\times \bbfR^N$ 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.