From the reviews: "…the book contains a wealth of material essential to the researcher concerned with multiple integral variational problems and with elliptic partial differential equations. The book not only reports the researches of the author but also the contributions of his contemporaries in the same and related fields. The book undoubtedly will become a standard reference for researchers in these areas. …The book is addressed mainly to mature mathematical analysts. However, any student of analysis will be greatly rewarded by a careful study of this book."
M. R. Hestenes in Journal of Optimization Theory and Applications
"The work intertwines in masterly fashion results of classical analysis, topology, and the theory of manifolds and thus presents a comprehensive treatise of the theory of multiple integral variational problems."
L. Schmetterer in Monatshefte für Mathematik
"The book is very clearly exposed and contains the last modern theory in this domain. A comprehensive bibliography ends the book."
M. Coroi-Nedeleu in Revue Roumaine de Mathématiques Pures et Appliquées
Author(s): Charles Bradfield Morrey Jr.
Series: Classics in Mathematics
Publisher: Springer Berlin Heidelberg
Year: 2008
Language: English
Pages: 522
Cover......Page 1
Series: Classics in Mathematics......Page 2
Reprint Title: Multiple Integrals in the Calculus of Variations......Page 4
Reprint Copyright......Page 5
Title: Multiple Integrals in the Calculus of Variations......Page 8
Copyright......Page 9
Preface......Page 10
Contents......Page 12
1.1. Introductory remarks......Page 17
1.2. The plan of the book: notation......Page 18
1.3. Very brief historical remarks......Page 21
1.4. The Euler equations......Page 23
1.5. Other classical necessary conditions......Page 26
1.6. Classical sufficient conditions......Page 28
1.7. The direct methods......Page 31
1.8. Lower semicontinuity......Page 35
1.9. Existence......Page 39
1.10. The differentiability theory. Introduction......Page 42
1.11. Differentiability; reduction to linear equations......Page 50
2.1. Introduction......Page 55
2.2. Elementary properties of harmonic functions......Page 56
2.3. Weyl's lemma......Page 57
2.4. Poisson's integral formula; elementary functions; Green's functions......Page 59
2.5. Potentials......Page 63
2.6. Generalized potential theory; singular integrals......Page 64
2.7. The Calderon-Zygmund inequalities......Page 71
2.8. The maximum principle for a linear elliptic equation of the second order......Page 77
3.1. Definitions and first theorems......Page 78
3.2. General boundary values; the spaces H^{m}_{p0}{G); weak convergence......Page 84
3.3. The Dirichlet problem......Page 86
3.4. Boundary values......Page 88
3.5. Examples; continuity; some Sobolev lemmas......Page 94
3.6. Miscellaneous additional results......Page 97
3.7. Potentials and quasi-potentials; generalizations......Page 102
4.1. The lower-semicontinuity theorems of Serrin......Page 106
4.2. Variational problems with f = f(p); the equations (1.10.13) with N = 1, B_i = 0, A^{a} = A^{a}(p)......Page 114
4.3. The borderline cases k = ν......Page 121
4.4. The general quasi-regular integral......Page 128
5.1. Introduction......Page 142
5.2. General theory; v > 2......Page 144
5.3. Extensions of the de Giorgi-Nash-Moser results; v > 2......Page 150
5.4. The case v = 2......Page 159
5.5. L_p and Schauder estimates......Page 165
5.6. The equation a\cdot\nabla^2 u + b\cdot\nabla u + cu -λu = f......Page 173
5.7. Analyticity of the solutions of analytic linear equations......Page 180
5.8. Analyticity of the solutions of analytic, non-linear, elliptic equations......Page 186
5.9. Properties of the extremals; regular cases......Page 202
5.10. The extremals in the case 1 < k < 2......Page 207
5.11. The theory of Ladyzenskaya and Ural'tseva......Page 210
5.12. A class of non-linear equations......Page 219
6.1. Introduction......Page 225
6.2. Interior estimates for general elliptic systems......Page 231
6.3. Estimates near the boundary; coerciveness......Page 241
6.4. Weak solutions......Page 258
6.5. The existence theory for the Dirichlet problem for strongly elliptic system......Page 267
6.6. The analyticity of the solutions of analytic systems of linear elliptic equations......Page 274
6.7. The analyticity of the solutions of analytic nonlinear elliptic systems......Page 282
6.8. The differentiability of the solutions of non-linear elliptic systems; weak solutions; a perturbation theorem......Page 293
7.1. Introduction......Page 302
7.2. Fundamentals; the Gaffney-Gårding inequality......Page 304
7.3. The variational method......Page 309
7.4. The decomposition theorem. Final results for compact manifolds without boundary......Page 311
7.5. Manifolds with boundary......Page 316
7.6. Differentiability at the boundary......Page 321
7.7. Potentials; the decomposition theorem......Page 325
7.8. Boundary value problems......Page 330
8.1. Introduction......Page 332
8.2. Results. Examples. The analytic embedding theorem......Page 336
8.3. Some important formulas......Page 344
8.4. The Hilbert space results......Page 349
8.5. The local analysis......Page 353
8.6. The smoothness results......Page 357
9.1. Introduction. Parametric integrals......Page 365
9.2. A lower semi-continuity theorem......Page 370
9.3. Two dimensional problems; introduction; the conformal mapping of surfaces......Page 378
9.4. The problem of Plateau......Page 390
9.5. The general two-dimensional parametric problem......Page 406
10.1. Introduction......Page 416
10.2. ν surfaces, their boundaries, and their Hausdorff measures......Page 423
10.3. The topological results of Adams......Page 430
10.4. The minimizing sequence; the minimizing set......Page 437
10.5. The local topological disc property......Page 455
10.6. The Reifenberg cone inequality......Page 475
10.7. The local differentiability......Page 490
10.8. Additional results of Federer concerning Lebesgue ν-area......Page 496
Bibliography......Page 510
Index......Page 520
