Direct methods in the calculus of variations / Enrico Giusti.

Format:

Book

Author/Creator:

Giusti, Enrico.

Language:

English

Subjects (All):

Calculus of variations.

Calculus.

Physical Description:

1 online resource (vii, 403 p.)

Edition:

1st ed.

Other Title:

Calculus of variations

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, c2003.

Language Note:

English

Summary:

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:

Introduction

ch. 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

ch. 2. Measurable functions. 2.1. L[symbol] 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

ch. 3. Sobolev spaces. 3.1. Partitions of unity. 3.2. Weak derivatives. 3.3. The Sobolev spaces W[symbol]. 3.4. Imbedding theorems. 3.5. Compactness. 3.6. Inequalities. 3.7. Traces. 3.8. The values of W[symbol] functions. 3.9. Notes and comments

ch. 4. Convexity and semicontinuity. 4.1. Preliminaries. 4.2. Convex functional. 4.3. Semicontinuity. 4.4. An existence theorem. 4.5. Notes and comments

ch. 5. Quasi-convex functional. 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

ch. 6. Quasi-minima. 6.1. Preliminaries. 6.2. Quasi-minima and differential quations. 6.3. Cubical quasi-minima. 6.4. L[symbol] estimates for the gradient. 6.5. Boundary estimates. 6.6. Notes and comments

ch. 7. Hölder 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

ch. 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. Hölder continuity of the derivatives (p = 2). 8.7. Other gradient estimates. 8.8. Hölder continuity of the derivatives (p ≠ 2). 8.9. Elliptic equations. 8.10. Notes and comments

ch. 9. Partial regularity. 9.1. Preliminaries. 9.2. Quadratic functionals. 9.3. The second Caccioppoli inequality. 9.4. The case F = F(z) (p = 2). 9.5. Partial regularity. 9.6. Notes and Comments

ch. 10. Higher derivatives. 10.1. Hilbert regularity. 10.2. Constant coefficients. 10.3. Continuous coefficients. 10.4. L[symbol] estimates. 10.5. Minima of functionals. 10.6. Notes and comments.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (p. 377-398) and index.

ISBN:

9786611935900

9781281935908

1281935905

9789812795557

9812795553