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**Direct methods in the calculus of variations.
(Metodi diretti nel calcolo delle variazioni.)**
*(Italian)*
Zbl 0942.49002

Bologna: Unione Matematica Italiana, v, 422 p. (1994).

This book is devoted to the central problem of the calculus of variations, that is the existence and the regularity of solutions for problems of the form Minimize \(J(u)=\int_\Omega F(x,u(x), Du(x))dx\), where \(\Omega\) is an open set in \({\mathbb R}^n\), \(u\) belongs to some class of functions defined on \(\Omega\), with values in \({\mathbb R}^m\), and \(F\) is defined on \(\Omega\times {\mathbb R}^n \times {\mathbb R}^{nm}\). The introductory chapter gives an overview of existence and semicontinuity results from Tonelli to Morrey. Next the main question investigated in the book, that is regularity results for solutions of the Euler-Lagrange equation of the above problem, is presented in detail. The fundamental result by De Giorgi, on Hölder continuity of solutions to elliptic equations in divergence form with discontinuous coefficients, is recalled. Counter-examples for systems led mathematicians to look for partial regularity results, that is regularity on an open set \(\Omega_0\) such that the Hausdorff dimension of \(\Omega\setminus \Omega_0\) is small. In this study, the functional analysis framework of Morrey spaces, Campanato spaces, and the techniques using quasi-minima have played a major role. All these developments are presented in a very nice way.

Chapter 1 is a short introduction to the maximum principle and to problems of minimum area. Chapter 2 is devoted to \(L^p\) spaces, Morrey spaces, Campanato spaces, the space of functions with Bounded Mean Oscillations, and their different imbeddings. Some interpolation results are also introduced. The study of function spaces continues in Chapter 3 with the Sobolev spaces. In particular compactness results, the Sobolev-Poincaré inequalities, trace theorems are presented in detail.

Existence theorems are stated in Chapters 4 and 5. Chapter 4 is concerned with functionals defined on spaces of scalar-valued functions (typically \(W^{1,p}(\Omega)\)). Semicontinuity and existence theorems are proved for functionals defined by convex normal integrands. The existence theory for functionals defined on spaces of vector valued functions is studied in Chapter 5. Existence and semicontinuity results are stated for quasiconvex integrands. Some elegant proofs are based on the Ekeland variational principle, which is presented in detail.

Chapter 6 is devoted to quasiminima, and to higher integrability results for the gradients of quasiminima. These questions are closely related to Hölder continuity results for solutions of elliptic systems. Scalar-valued minimizers or quasiminima of functionals are studied in Chapter 7. The De Giorgi class is introduced. Hölder regularity results up to the boundary are proven. Hölder continuity for the gradient of minimizers for functionals whose behavior is close to \(\int_\Omega |Du|^pdx\), are obtained in Chapter 8.

Chapter 9 is devoted to partial regularity, typically Hölder continuity is expected on an open set \(\Omega_0\subset \Omega\), such that the Hausdorff dimension of \(\Omega\setminus \Omega_0\) is strictly less than \(n-2\). Chapter 10 is concerned with some additional regularity results for linear elliptic systems.

The reading of the book is very pleasant. Many comments throughout the different chapters help the reader. At the end of each chapter, additional comments give an overview of complementary results. An index and an extensive bibliography enrich this pleasant book, which can be considered as a valuable update of the one by M. Giaquinta on the same subject [“Multiple integrals in the calculus of variations and nonlinear elliptic systems” (1983; Zbl 0516.49003)].

Chapter 1 is a short introduction to the maximum principle and to problems of minimum area. Chapter 2 is devoted to \(L^p\) spaces, Morrey spaces, Campanato spaces, the space of functions with Bounded Mean Oscillations, and their different imbeddings. Some interpolation results are also introduced. The study of function spaces continues in Chapter 3 with the Sobolev spaces. In particular compactness results, the Sobolev-Poincaré inequalities, trace theorems are presented in detail.

Existence theorems are stated in Chapters 4 and 5. Chapter 4 is concerned with functionals defined on spaces of scalar-valued functions (typically \(W^{1,p}(\Omega)\)). Semicontinuity and existence theorems are proved for functionals defined by convex normal integrands. The existence theory for functionals defined on spaces of vector valued functions is studied in Chapter 5. Existence and semicontinuity results are stated for quasiconvex integrands. Some elegant proofs are based on the Ekeland variational principle, which is presented in detail.

Chapter 6 is devoted to quasiminima, and to higher integrability results for the gradients of quasiminima. These questions are closely related to Hölder continuity results for solutions of elliptic systems. Scalar-valued minimizers or quasiminima of functionals are studied in Chapter 7. The De Giorgi class is introduced. Hölder regularity results up to the boundary are proven. Hölder continuity for the gradient of minimizers for functionals whose behavior is close to \(\int_\Omega |Du|^pdx\), are obtained in Chapter 8.

Chapter 9 is devoted to partial regularity, typically Hölder continuity is expected on an open set \(\Omega_0\subset \Omega\), such that the Hausdorff dimension of \(\Omega\setminus \Omega_0\) is strictly less than \(n-2\). Chapter 10 is concerned with some additional regularity results for linear elliptic systems.

The reading of the book is very pleasant. Many comments throughout the different chapters help the reader. At the end of each chapter, additional comments give an overview of complementary results. An index and an extensive bibliography enrich this pleasant book, which can be considered as a valuable update of the one by M. Giaquinta on the same subject [“Multiple integrals in the calculus of variations and nonlinear elliptic systems” (1983; Zbl 0516.49003)].

Reviewer: J.-P.Raymond (Toulouse)

### MSC:

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

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

49J10 | Existence theories for free problems in two or more independent variables |

49N60 | Regularity of solutions in optimal control |

35J50 | Variational methods for elliptic systems |