Direct methods in the calculus of variations
著者
書誌事項
Direct methods in the calculus of variations
World Scientific, c2003
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注記
Includes bibliographical references (p. 377-398) and index
内容説明・目次
内容説明
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.
目次
- Semi-Classical Theory
- Integrable Functions
- Sobolev Spaces
- Semicontinuity
- Quasi-Convex Functionals
- Quasi-Minima
- Regularity of Quasi-Minima
- First Derivatives
- Partial Regularity
- Higher Derivatives.
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