Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory.

*(English)*Zbl 1255.49074
Cambridge Studies in Advanced Mathematics 135. Cambridge: Cambridge University Press (ISBN 978-1-107-02103-7/hbk; 978-1-139-10813-3/ebook). xix, 454 p. (2012).

This book originates from the lecture notes that the author held at the University of Duisburg-Essen in 2005. The first aim of the book was to provide an introduction for the beginners about theory of sets of finite perimeter, presenting some results concerning the existence, symmetry, regularity and structure of singularities in some variational problems involving the length and area. Topics like the Euclidean isoperimetric problem, the description of geometric properties of equilibrium shapes for liquid drops and crystals, the regularity up to a singular set of codimension at least \(8\) for area minimizing boundaries, and the theory of minimizing clusters
are covered. The secondary aim of this book is to provide a multi-leveled introduction to the study of other variational problems (parametric and non-parametric) as well as of partial differential equations. In this way, an interested reader is able to enter with relative ease several parts of Geometric Measure Theory and to apply some tools from this theory in the study of other problems from Mathematics.

The book is divided in four parts. Part I Radon measures on \(\mathbb{R}^n\), contains the basic theory of Radon measures, Hausdorff measures and rectifiable sets and provides the background material for the rest of the book. The author tried to develop this part as independent, self-contained, and easily accessible reading. In the first six chapters, the author deals with some more abstract aspects of the basic theory of Radon measures on \(\mathbb{R}^n\). Next, the differentiation theory and its applications are discussed. The Radon measures are considered from a geometric viewpoint, focusing on the interaction between Euclidean geometry and Measure theory, and covering topics such as Lipschitz functions, Hausdorff measures, area formulas, rectifiable sets and measure-theoretic differentiability. In the Part II of the book, the Sets of finite perimeter, are studied. A generalization of the Gauss-Green theorem based on the notion of vector valued Radon measure is presented. The link with the theory of Radon measures is exploited to deduce some basic lower semicontinuity and compactness theorems for sequences of locally finite perimeter. Then the possibility of approximating sets of finite perimeter by sequences of open sets with smooth boundary is discussed. In chapter 14, the author studies the Euclidean isoperimetric problem. The Euclidean balls are characterized as the unique minimizers in the Euclidean isoperimetric problem. In chapter 15, the De Giorgi structure theory is presented. In Chapter 16, the study of reduced boundaries and Gauss-Green measures is made by using the theory of rectifiable sets developed earlier, and the Federer theorem about the \(\mathcal{H}_{n-1}\)-equivalence between the reduced boundary of a set \(E\) of finite perimeter, the set \(E^{(1/2)}\) of its point of density one-half, and the essential boundary \(\partial^c E\) of \(E\). Next the author applies the area formula to study the behavior of sets of finite perimeter under the action of one parameter families of diffeomorphisms. He computes the first and second variation formulas of perimeter. Then, he discusses, in chapter 18, some aspect concerning the co-area formula, mainly the slicing of reduced boundaries. In chapters 19 and 20, the author discusses the equilibrium problem for a liquid confined inside a given container and anisotropic surface energy, considering the Wulff problem. In the Part III the Regularity theory and analysis of singularitiesis discussed. The main goal of this part is the proof of a theorem concerning the structure of \(A\cap \partial^*E\) where \(A\) is an open set in \(\mathbb{R}^n\), and E is a local perimeter minimizer in \(A\). Some properties of the singular set of \(E\) in \(A\) are obtained. The author introduces \((\Lambda ,r_0)\)-perimeter minimality and the fundamental notion of excess \(\mathbf{e}(E,x,r)\). Then he uses the reverse Poincaré inequality and some basic properties of harmonic functions to prove some explicit decay estimates of \(\nabla u\). A connection with elliptic equations in divergence form is established and the singular sets and singular minimizing cones are studied. In the part IV, Minimizing clusters , are studied. The Almgren theorem concerning the existence and everywhere regularity of minimizers in the partitioning problem is studied and proved. The planar minimizing clusters, the structure of singularities in higher dimensions, symmetry properties of minimizing clusters, the double bubble theorem and the multi-phase anisotropic partition problem are described.

This is a well written book by a specialist in the field of Geometric Measure Theory. It provides a generous guidance to the reader. The book is recommended, mainly, not only to the beginners who can find a brought up-to-date source in the field but also to specialists who can find some genuine aspects from the theory. It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.

The book is divided in four parts. Part I Radon measures on \(\mathbb{R}^n\), contains the basic theory of Radon measures, Hausdorff measures and rectifiable sets and provides the background material for the rest of the book. The author tried to develop this part as independent, self-contained, and easily accessible reading. In the first six chapters, the author deals with some more abstract aspects of the basic theory of Radon measures on \(\mathbb{R}^n\). Next, the differentiation theory and its applications are discussed. The Radon measures are considered from a geometric viewpoint, focusing on the interaction between Euclidean geometry and Measure theory, and covering topics such as Lipschitz functions, Hausdorff measures, area formulas, rectifiable sets and measure-theoretic differentiability. In the Part II of the book, the Sets of finite perimeter, are studied. A generalization of the Gauss-Green theorem based on the notion of vector valued Radon measure is presented. The link with the theory of Radon measures is exploited to deduce some basic lower semicontinuity and compactness theorems for sequences of locally finite perimeter. Then the possibility of approximating sets of finite perimeter by sequences of open sets with smooth boundary is discussed. In chapter 14, the author studies the Euclidean isoperimetric problem. The Euclidean balls are characterized as the unique minimizers in the Euclidean isoperimetric problem. In chapter 15, the De Giorgi structure theory is presented. In Chapter 16, the study of reduced boundaries and Gauss-Green measures is made by using the theory of rectifiable sets developed earlier, and the Federer theorem about the \(\mathcal{H}_{n-1}\)-equivalence between the reduced boundary of a set \(E\) of finite perimeter, the set \(E^{(1/2)}\) of its point of density one-half, and the essential boundary \(\partial^c E\) of \(E\). Next the author applies the area formula to study the behavior of sets of finite perimeter under the action of one parameter families of diffeomorphisms. He computes the first and second variation formulas of perimeter. Then, he discusses, in chapter 18, some aspect concerning the co-area formula, mainly the slicing of reduced boundaries. In chapters 19 and 20, the author discusses the equilibrium problem for a liquid confined inside a given container and anisotropic surface energy, considering the Wulff problem. In the Part III the Regularity theory and analysis of singularitiesis discussed. The main goal of this part is the proof of a theorem concerning the structure of \(A\cap \partial^*E\) where \(A\) is an open set in \(\mathbb{R}^n\), and E is a local perimeter minimizer in \(A\). Some properties of the singular set of \(E\) in \(A\) are obtained. The author introduces \((\Lambda ,r_0)\)-perimeter minimality and the fundamental notion of excess \(\mathbf{e}(E,x,r)\). Then he uses the reverse Poincaré inequality and some basic properties of harmonic functions to prove some explicit decay estimates of \(\nabla u\). A connection with elliptic equations in divergence form is established and the singular sets and singular minimizing cones are studied. In the part IV, Minimizing clusters , are studied. The Almgren theorem concerning the existence and everywhere regularity of minimizers in the partitioning problem is studied and proved. The planar minimizing clusters, the structure of singularities in higher dimensions, symmetry properties of minimizing clusters, the double bubble theorem and the multi-phase anisotropic partition problem are described.

This is a well written book by a specialist in the field of Geometric Measure Theory. It provides a generous guidance to the reader. The book is recommended, mainly, not only to the beginners who can find a brought up-to-date source in the field but also to specialists who can find some genuine aspects from the theory. It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.

Reviewer: Vasile Oproiu (Iaşi)

##### MSC:

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

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

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

28A75 | Length, area, volume, other geometric measure theory |

58E30 | Variational principles in infinite-dimensional spaces |