##
**Variational principles in mathematical physics, geometry, and economics. Qualitative analysis of nonlinear equations and unilateral problems.**
*(English)*
Zbl 1206.49002

Encyclopedia of Mathematics and Its Applications 136. Cambridge: Cambridge University Press (ISBN 978-0-521-11782-1/hbk). xv, 368 p. (2010).

The use of variational principles has a long and fruitful history in mathematics and physics, both in solving problems and shaping theories, and, recently, it has been introduced in economics. The book consists of three parts.

Part I consists of eight chapters and is devoted to variational principles in mathematical physics. Chapter 1 briefly recalls the main variational principles which will be used in the rest of the book, such as Ekeland’s and Borwein-Preiss’s variational principles, minimax- and minimization-type principles (the mountain pass theorem, Ricceri-type multiplicity theorems, the Brezis-Nirenberg minimization technique), the principle of symmetric criticality for non-smooth Szulkin-type functionals, as well as Pohozaev’s fibering method. Chapter 2 treats variational inequalities on unbounded strips and for area-type functionals related to convex energy functionals and non-convex problems for the potential. In chapter 3, the authors are concerned with the study of certain quasi-linear eigenvalue problem in weighted Sobolev spaces. Chapters 4 through 6 are devoted to a substantial study of systems of elliptic partial differential equations. The authors examine some basic facts about the theory of boundary value problems for elliptic systems of partial differential equations, where they are mainly concerned with the existence of and multiplicity results for eigenvalue problems involving elliptic systems of gradient type. Some systems with arbitrary growth nonlinearities and scalar field systems are also studied. These are challenging topics of growing importance, with many applications in natural and human sciences, such as demography. The purpose of chapter 7 is to study the number and the behavior of solutions to a Dirichlet problem which involves an oscillatory nonlinearity and a pure power term, as well as the effect of an arbitrary perturbation in a constrained nonlinear problem with symmetry. Part I ends up with the selection of problems forming chapter 8.

Part II consists of four chapters and demonstrates the importance of variational problems in geometry. Chapter 9 deals with elliptic problems defined on compact Riemannian manifolds. Classical questions concerning geodesics or minimal surfaces are not considered, but instead the authors concentrate on a less standard problem, namely the transformation of classical questions related to the Emden-Fowler equation into problems defined on some four dimensional sphere. In Chapter 10, the authors study a nonlinear elliptic problem defined on the unit sphere, involving an asymptotically critical oscillatory term and the Laplace-Beltrami operator. By a group-theoretical argument, the existence of infinitely many sign-changing solutions for the studied problem are guaranteed, giving also a lower estimate of the number of those sequences of solutions whose elements in different sequences are mutually symmetrically distinct. Chapter 11 treats elliptic equations with critical exponent defined on compact Riemannian manifolds. This problem is closely related to the so-called Yamabe problem, that is, for any smooth compact Riemannian manifold \((M, g)\) there exists a conformal metric \(\widetilde g\) to \(g\) with constant scalar curvature. In Chapter 12, there are problems related to Part II. The last part consists of five chapters and deals with variational principles in economics. After some mathematical introduction in Chapter 13, the authors study the minimization of cost-functions on manifolds in Chapters 14 and 15, giving special attention to the Finslerian-Poincaré disc and best approximation problems on manifolds. Then, in Chapter 16, they consider a variational approach of Nash equilibria through variational inequalities.

Some problems related to Part III are found in Chapter 17. Five appendices illustrate some basic mathematical tools applied in this book: elements of convex analysis, function spaces, category and genus, Clarke and Degiovanni gradients, and elements of set-valued analysis. These auxiliary chapters deal with some analytical methods used in this volume, but also include some complements.

The interesting method of presentation of the book, with extensive reference list and index, make me believe that the book will be appreciated by mathematicians, engineers, economists, physicists, and all scientists interested in variational methods and in their applications.

Part I consists of eight chapters and is devoted to variational principles in mathematical physics. Chapter 1 briefly recalls the main variational principles which will be used in the rest of the book, such as Ekeland’s and Borwein-Preiss’s variational principles, minimax- and minimization-type principles (the mountain pass theorem, Ricceri-type multiplicity theorems, the Brezis-Nirenberg minimization technique), the principle of symmetric criticality for non-smooth Szulkin-type functionals, as well as Pohozaev’s fibering method. Chapter 2 treats variational inequalities on unbounded strips and for area-type functionals related to convex energy functionals and non-convex problems for the potential. In chapter 3, the authors are concerned with the study of certain quasi-linear eigenvalue problem in weighted Sobolev spaces. Chapters 4 through 6 are devoted to a substantial study of systems of elliptic partial differential equations. The authors examine some basic facts about the theory of boundary value problems for elliptic systems of partial differential equations, where they are mainly concerned with the existence of and multiplicity results for eigenvalue problems involving elliptic systems of gradient type. Some systems with arbitrary growth nonlinearities and scalar field systems are also studied. These are challenging topics of growing importance, with many applications in natural and human sciences, such as demography. The purpose of chapter 7 is to study the number and the behavior of solutions to a Dirichlet problem which involves an oscillatory nonlinearity and a pure power term, as well as the effect of an arbitrary perturbation in a constrained nonlinear problem with symmetry. Part I ends up with the selection of problems forming chapter 8.

Part II consists of four chapters and demonstrates the importance of variational problems in geometry. Chapter 9 deals with elliptic problems defined on compact Riemannian manifolds. Classical questions concerning geodesics or minimal surfaces are not considered, but instead the authors concentrate on a less standard problem, namely the transformation of classical questions related to the Emden-Fowler equation into problems defined on some four dimensional sphere. In Chapter 10, the authors study a nonlinear elliptic problem defined on the unit sphere, involving an asymptotically critical oscillatory term and the Laplace-Beltrami operator. By a group-theoretical argument, the existence of infinitely many sign-changing solutions for the studied problem are guaranteed, giving also a lower estimate of the number of those sequences of solutions whose elements in different sequences are mutually symmetrically distinct. Chapter 11 treats elliptic equations with critical exponent defined on compact Riemannian manifolds. This problem is closely related to the so-called Yamabe problem, that is, for any smooth compact Riemannian manifold \((M, g)\) there exists a conformal metric \(\widetilde g\) to \(g\) with constant scalar curvature. In Chapter 12, there are problems related to Part II. The last part consists of five chapters and deals with variational principles in economics. After some mathematical introduction in Chapter 13, the authors study the minimization of cost-functions on manifolds in Chapters 14 and 15, giving special attention to the Finslerian-Poincaré disc and best approximation problems on manifolds. Then, in Chapter 16, they consider a variational approach of Nash equilibria through variational inequalities.

Some problems related to Part III are found in Chapter 17. Five appendices illustrate some basic mathematical tools applied in this book: elements of convex analysis, function spaces, category and genus, Clarke and Degiovanni gradients, and elements of set-valued analysis. These auxiliary chapters deal with some analytical methods used in this volume, but also include some complements.

The interesting method of presentation of the book, with extensive reference list and index, make me believe that the book will be appreciated by mathematicians, engineers, economists, physicists, and all scientists interested in variational methods and in their applications.

Reviewer: Andrew Bucki (Edmond)

### MSC:

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

35A15 | Variational methods applied to PDEs |

49J40 | Variational inequalities |

49K35 | Optimality conditions for minimax problems |

49S05 | Variational principles of physics |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

91B02 | Fundamental topics (basic mathematics, methodology; applicable to economics in general) |