Direct methods in the calculus of variations

“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.”
Zentralblatt MATH
“It can be recommended for graduate courses or post-graduate courses in the calculus of variations, or as reference text.”
Studia Universitatis Babes-Bolyai, Series Mathematica
“The exposition is always clear and self-contained … therefore this book may serve well as a textbook for a graduate course on the subject. Each chapter is complemented with detailed historical notes and interesting results which may be difficult to find elsewhere.”
Mathematical Reviews
 
This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.