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Direct Methods In The Calculus Of Variations - Enrico Giusti

Direct Methods In The Calculus Of Variations

Hardcover Published: 24th January 2003
ISBN: 9789812380432
Number Of Pages: 412

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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 were widely-used in the 20th 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 work, 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 volume is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.

.,." complemented with detailed historical notes and interesting results which may be difficult to find elsewhere.?

Introductionp. 1
Semi-Classical Theoryp. 13
The Maximum Principlep. 14
The Bounded Slope Conditionp. 18
Barriersp. 22
The Area Functionalp. 30
Non-Existence of Minimal Surfacesp. 32
Measurable Functionsp. 39
L[superscript p] Spacesp. 39
Test Functions and Mollifiersp. 44
Morrey's and Campanato's Spacesp. 46
The Lemmas of John and Nirenbergp. 54
Interpolationp. 61
The Hausdorff Measurep. 68
Sobolev Spacesp. 75
Partitions of Unityp. 75
Weak Derivativesp. 79
The Sobolev Spaces W[superscript k,p]p. 83
Imbedding Theoremsp. 90
Compactnessp. 98
Inequalitiesp. 101
Tracesp. 106
The Values of W[superscript 1,p] Functionsp. 110
Convexity and Semicontinuityp. 119
Preliminariesp. 119
Convex Functionalsp. 121
Semicontinuityp. 123
An Existence Theoremp. 131
Quasi-Convex Functionalsp. 139
Necessary Conditionsp. 139
First Semicontinuity Resultsp. 150
The Quasi-Convex Envelopep. 155
The Ekeland Variational Principlep. 160
Semicontinuityp. 162
Coerciveness and Existencep. 168
Quasi-Minimap. 173
Preliminariesp. 173
Quasi-Minima and Differential Quationsp. 175
Cubical Quasi-Minimap. 187
L[superscript p] Estimates for the Gradientp. 197
Boundary Estimatesp. 206
Holder Continuityp. 213
Caccioppoli's Inequalityp. 213
De Giorgi Classesp. 218
Quasi-Minimap. 225
Boundary Regularityp. 232
The Harnack Inequalityp. 235
The Homogeneous Casep. 242
[omega]-Minimap. 245
Boundary Regularityp. 255
First Derivativesp. 261
The Difference Quotientsp. 263
Second Derivativesp. 266
Gradient Estimatesp. 271
Boundary Estimatesp. 274
[omega]-Minimap. 278
Holder Continuity of the Derivatives (p = 2)p. 285
Other Gradient Estimatesp. 288
Holder Continuity of the Derivatives p [is not equal to] 2p. 298
Elliptic Equationsp. 301
Partial Regularityp. 307
Preliminariesp. 307
Quadratic Functionalsp. 309
The Second Caccioppoli Inequalityp. 319
The Case F = F(z) (p = 2)p. 329
Partial Regularityp. 333
Higher Derivativesp. 347
Hilbert Regularityp. 348
Constant Coefficientsp. 355
Continuous Coefficientsp. 362
L[superscript p] Estimatesp. 368
Minima of Functionalsp. 374
Referencesp. 377
Indexp. 399
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9789812380432
ISBN-10: 9812380434
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 412
Published: 24th January 2003
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.88  x 2.54
Weight (kg): 0.72