Direct methods in the calculus of variations. (English) Zbl 1028.49001

Giusti, Enrico

Singapore: World Scientific. vii, 403 p. (2003).

This is a very nice and self-contained book on the calculus of variations. The main two topics of the book are the existence theory and the regularity of minima. To include a large class of problems the regularity results are stated for quasi-minima and \(\omega\)-minima. In Chapter 1, as an introduction to the subject, the author studies scalar functionals depending only on the gradient. The existence of minima for the Dirichlet problem is proved in the space of Lipschitz continuous functions. The results are obtained by means of elementary techniques. The space of measurable functions, the \(L^p\) spaces, Lorentz spaces, Campanato spaces, and Morrey spaces are introduced in a concise way in Chapter 2. The Sobolev spaces are studied in Chapter 3. The main semicontinuity results based on convexity and quasi-convexity of integrands are proved in the two following chapters. The notion of quasi-minimum and its relationships with solutions of elliptic equations and systems is studied in Chapter 6. The seventh and eighth chapters are devoted to the Hölder regularity of scalar quasi-minima, of \(\omega\)-minima of functionals, and of solutions of nonlinear elliptic equations in divergence form. The partial regularity of \(\omega\)-minima of quasi-convex functionals is treated in Chapter 9, and the last two chapters are concerned with the regularity of higher derivatives of solutions of elliptic equations. Notes and comments at the end of each chapter as well as the introductory chapter give some enlightening explanations on the historic developments of concepts and notions. This book must be recommended both to beginners in variational calculus and to more confirmed specialists in regularity theory of elliptic problems. It will become a reference in the calculus of variations and it contains in one volume of a reasonable size a very clear presentation of deep results.

Reviewer: Jean-Pierre Raymond (Toulouse)

MSC:

49-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49J10	Existence theories for free problems in two or more independent variables
49J45	Methods involving semicontinuity and convergence; relaxation
49N60	Regularity of solutions in optimal control
35B65	Smoothness and regularity of solutions to PDEs
35Jxx	Elliptic equations and elliptic systems

Keywords:

calculus of variations; Sobolev spaces; Lorentz spaces; Morrey spaces; Campanato spaces; convexity; quasi-convexity; regularity; partial regularity