Direct Methods in the Calculus of Variations.
Title:
Direct Methods in the Calculus of Variations.
Author:
Giusti, Enrico.
ISBN:
9789812795557
Personal Author:
Physical Description:
1 online resource (412 pages)
Contents:
Contents -- Introduction -- Chapter 1 Semi-Classical Theory -- 1.1 The Maximum Principle -- 1.2 The Bounded Slope Condition -- 1.3 Barriers -- 1.4 The Area Functional -- 1.5 Non-Existence of Minimal Surfaces -- 1.6 Notes and Comments -- Chapter 2 Measurable Functions -- 2.1 Lp Spaces -- 2.2 Test Functions and Mollifiers -- 2.3 Morrey's and Campanato's Spaces -- 2.4 The Lemmas of John and Nirenberg -- 2.5 Interpolation -- 2.6 The Hausdorff Measure -- 2.7 Notes and Comments -- Chapter 3 Sobolev Spaces -- 3.1 Partitions of Unity -- 3.2 Weak Derivatives -- 3.3 The Sobolev Spaces Wkp -- 3.4 Imbedding Theorems -- 3.5 Compactness -- 3.6 Inequalities -- 3.7 Traces -- 3.8 The Values of W1p Functions -- 3.9 Notes and Comments -- Chapter 4 Convexity and Semicontinuity -- 4.1 Preliminaries -- 4.2 Convex Functional -- 4.3 Semicontinuity -- 4.4 An Existence Theorem -- 4.5 Notes and Comments -- Chapter 5 Quasi-Convex Functionals -- 5.1 Necessary Conditions -- 5.2 First Semicontinuity Results -- 5.3 The Quasi-Convex Envelope -- 5.4 The Ekeland Variational Principle -- 5.5 Semicontinuity -- 5.6 Coerciveness and Existence -- 5.7 Notes and Comments -- Chapter 6 Quasi-Minima -- 6.1 Preliminaries -- 6.2 Quasi-Minima and Differential Quations -- 6.3 Cubical Quasi-Minima -- 6.4 Lp Estimates for the Gradient -- 6.5 Boundary Estimates -- 6.6 Notes and Comments -- Chapter 7 Holder Continuity -- 7.1 Caccioppoli's Inequality -- 7.2 De Giorgi Classes -- 7.3 Quasi-Minima -- 7.4 Boundary Regularity -- 7.5 The Harnack Inequality -- 7.6 The Homogeneous Case -- 7.7 w-Minima -- 7.8 Boundary Regularity -- 7.9 Notes and Comments -- Chapter 8 First Derivatives -- 8.1 The Difference Quotients -- 8.2 Second Derivatives -- 8.3 Gradient Estimates -- 8.4 Boundary Estimates -- 8.5 w-Minima.
8.6 Holder Continuity of the Derivatives (p=2) -- 8.7 Other Gradient Estimates -- 8.8 Holder Continuity of the Derivatives (p#2) -- 8.9 Elliptic Equations -- 8.10 Notes and Comments -- Chapter 9 Partial Regularity -- 9.1 Preliminaries -- 9.2 Quadratic Functional -- 9.3 The Second Caccioppoli Inequality -- 9.4 The Case F = F(z) (p = 2) -- 9.5 Partial Regularity -- 9.6 Notes and Comments -- Chapter 10 Higher Derivatives -- 10.1 Hilbert Regularity -- 10.2 Constant Coefficients -- 10.3 Continuous Coefficients -- 10.4 Lp Estimates -- 10.5 Minima of Functionals -- 10.6 Notes and Comments -- References -- Index.
Abstract:
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. Contents: Semi-Classical Theory; Measurable Functions; Sobolev Spaces; Convexity and Semicontinuity; Quasi-Convex Functionals; Quasi-Minima; Hölder Continuity; First Derivatives; Partial Regularity; Higher Derivatives. Readership: Graduate students, academics and researchers in the field of analysis and differential equations.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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