# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Direct methods in the calculus of variations. (English) Zbl 1028.49001
Singapore: World Scientific. vii, 403 p. £33.00 (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.
##### MSC:
 49-01 Textbooks (calculus of variations) 49J10 Free problems in several independent variables (existence) 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 49N60 Regularity of solutions in calculus of variations 35B65 Smoothness and regularity of solutions of PDE 35Jxx Elliptic equations and systems