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Direct methods in the calculus of variations

Author: Enrico Giusti
Publisher: Singapore ; River Edge, NJ : World Scientific, ©2003.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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  Read more...
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Enrico Giusti
OCLC Number: 646768574
Description: 1 online resource (vii, 403 pages)
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.
Other Titles: Calculus of variations
Responsibility: Enrico Giusti.

Abstract:

A self-contained 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  Read more...

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