Mathematics in Science and Engineering

PREFACE 3
Contents 11
1. INTRODUCTION 15
1. The Basic Notation and Terminology 15
2. Admissible Extensions of the Concept of Solution of Linear and Quasilinear Equations 23
3. The Basic Results and Their Possible Development 41
2. AUXILIARY PROPOSITIONS 53
1. Some Simple Inequalities 53
2. Functions in the Class Vlm(O) 55
3. Functions in the Classes Wlm(O) and Wlm(O) 64
4. Other Auxiliary Propositions 70
5. Bounds for max|u(x)| 84
6. The Function Class B(O,M,g,d,1/q) 95
7. The Classes B(O+S_1,...) and B_m(O+S_1,...) 104
8. The Classes 109
3. LINEAR EQUATIONS 120
1. Solvability of the Dirichlet Problem in the Spaces Cla(O). l>2 121
2. Schauder’s A Priori Estimate 130
3. The Solvability in C2,a(O) of Other Boundary-Value Problems 148
4. Generalized Solutions in W12(O). The First Fundamental Inequality 152
5. Solvability of the First Boundary-Value Problem in (Q) 163
6. The Second and Third Boundary-Value Problems 174
7. Interior Estimates in Lt of the Second Derivatives of an Arbitrary Function in Terms of the Values of an Elliptic Operator Applied to It 176
8. The Second Fundamental Inequality for Elliptic Operators 183
9. On the Solvability of the First Boundary-Value Problem in the Space 197
10. Conditions Under Which Generalized Solutions in U^(2) Belong to 199
11. Other Ways of Proving the Second Fundamental Inequality 204
12. Conditions Under Which Generalized Solutions in W22 Belong to Cla for l>2 207
13. The Boundedness of Genera lized Solutions in 209
14. Conditions Under Which Generalized Solutions in Belong to Co 215
15. Conditions for Boundedness of max| Va| and |uxl| for Generalized Solutions in W12(O) 217
16. Diffraction Problems 219
17. The Case of Two Independent Variables 237
18. Two-Dimensional Saddle-Shaped Surfaces 252
4. QUASILINEAR EQUATIONS WITH PRINCIPAL PART IN DIVERGENCE FORM 258
1. Bounded Generalized Solutions. Continuity in the Sense of Holder 259
2. Uniqueness in the Small 268
3. A Bound for max|u| 272
4. A Bound for max|Vu| Over the Entire Region O 280
5. Generalized Second-Order Derivatives 284
6. A Bound for the Norms |u|la, where I>1 291
7. Bounds for the Maximum Absolute Values of Generalized Solutions 299
8. Existence Theorems 305
5. VARIATIONAL PROBLEMS 332
1. Statement of the Problems 332
2. Finding Functions That Minimize the Functional I(U) 335
3. Finding a Bound for the Maximum Absolute Value of Solutions of Variational Problems 341
4. Proof of Holderness of Generalized Solutions 345
5. The Theorem on Uniqueness in the Small of Generalized Solutions 348
6. Further Investigation of the Differentiability Properties of Generalized Solutions 349
6. QUASILINEAR EQUATIONS OF THE GENERAL FORM 352
1. A Bound for the Norm |u||a, in terms of the Norm Iuli,o 353
2. A Bound for max|Vu| 363
3. Existence Theorems 384
4. Two-Dimensional Problems 393
7. LINEAR SYSTEMS OF ELLIPTIC EQUATIONS 400
1. Generalized Solutions in W12(O) 400
2. A Bound for max|u| 403
3. A Bound for |u|ao 409
4. A Priori Bounds for |u|la0 and ||u||W22(O) 412
5. Solvability of the Problem (1.1), (1.3) in the Classes Cla(O) 416
6. Differentiability Properties of Generalized Solutions 418
8. QUASILINEAR SYSTEMS 420
1. A Priori Bounds for the Norms |u|laO, where l>1 in Terms of max|u, Vu I 421
2. A Bound for |u|a,O 423
3. The Energy Inequality and a Bound for max|vu| on the Boundary 427
4. A Bound for max|Vu|| 431
5. Existence Theorems 434
9. OTHER DEVICES FOR OBTAINING BOUNDS FOR THE HOLDER CONSTANTS FOR SOLUTIONS AND THEIR DERIVATIVES 437
1. The Case of the Simplest Equation 439
2. Bounds on Holder Constants for Solutions of Equations (Linear and Quasilinear) with Principal Part in Divergence Form 443
3. Bounds on the Oscillations of the Derivatives of Solutions of Equations with Principal Part in Divergence Form 456
4. Equations Not in Divergence Form 457
5. Moser’s Method of Bounding |u|aO for Solutions of Linear Equations 460
6. Nirenberg’s Estimate 466
7. Morrey’s Method for Finding a Bound for the Holder Constant for Solutions of Two-Dimensional Variational Problems 470
10. OTHER BOUNDARY-VALUE PROBLEMS 473
1. Formulation of the Problems and the General Procedure for Solving Them 473
2. Quasilinear Equations with Principal Part in Divergence Form 479
3. Quasilinear Systems 495
BIBLIOGRAPHY 498
INDEX 507