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Product Details
ISBN-13: | 9789814452328 |
---|---|
Publisher: | World Scientific Publishing Company, Incorporated |
Publication date: | 05/29/2013 |
Pages: | 528 |
Product dimensions: | 6.10(w) x 8.90(h) x 1.30(d) |
Table of Contents
Preface vii
1 Pointwise Estimates 1
1.1 The maximum principle 3
1.2 The definition of obliqueness 6
1.3 The case c < 0, β0 ≤ 0 7
1.4 A generalized change of variables formula 11
1.5 The Aleksandrov-Bakel'man-Pucci maximum principles 13
1.6 The interior weak Harnack inequality 18
1.7 The weak Harnack inequality at the boundary 21
1.8 The strong maximum principle and uniqueness 29
1.9 Hölder continuity 31
1.10 The local maximum principle 34
1.11 Pointwise estimates for solutions of mixed boundary value problems 35
1.12 Derivative bounds for solutions of elliptic equations 38
2 Classical Schauder Theory from a Modern Perspective 47
2.1 Definitions and properties of Hölder spaces 47
2.2 An alternative characterization of Hölder spaces 58
2.3 An existence result 60
2.4 Basic interior estimates 61
2.5 The Perron process for the Dirichlet problem 65
2.6 A model mixed boundary value problem 71
2.7 Domains with curved boundary 81
2.8 Fredholm-Riesz-Schauder theory 83
3 The Miller Barrier and Some Supersolutions for Oblique Derivative Problems 89
3.1 Theory of ordinary differential equations 89
3.2 The Miller barrier construction 95
3.3 Construction of supersolutions for Dirichlet data 102
3.4 Construction of a supersolution for oblique derivative problems 105
3.5 The strong maximum principle, revisited 109
3.6 A Miller barrier for mixed boundary value problems 111
4 Hölder Estimates for First and Second Derivatives 117
4.1 C1,α estimates for continuous β 117
4.2 Regularized distance 129
4.3 Existence of solutions for continuous β 131
4.4 Hölder gradient estimates for the Dirichlet problem 132
4.5 C1,α estimates with discontinuous β in two dimensions 136
4.6 C1,α estimates for discontinuous β in higher dimensions 159
4.7 C2,α estimates 165
5 Weak Solutions 171
5.1 Definitions and basic properties of weak derivatives 173
5.2 Sobolev imbedding theorems 174
5.3 Poincaré's inequality 177
5.4 The weak maximum principle 180
5.5 Trace theorems 182
5.6 Existence of weak solutions 187
5.7 Higher regularity of solutios 190
. Regularity of solutions 190
5.8 Global boundedness of weak solutions 193
5.9 The local maximum principle 201
5.10 The DeGiorgi class 203
5.11 Membership of supersolutions in the De Giorgi class 208
5.12 Consequences of the local estimates 210
5.13 Integral characterizations of Hölder spaces 211
5.14 Schauder estimates 214
6 Strong Solutions 227
6.1 Pointwise estimates for strong solutions 227
6.2 A sharp trace theorem 231
6.3 Results from harmonic analysis 235
6.4 Some further estimates for boundary value problems in a spherical cap 238
6.5 Lp estimates for solutions of constant coefficient problems in a spherical cap 241
6.6 Local estimates for strong solutions of constant coefficient problems 246
6.7 Local interior Lp estimates for the second derivatives of strong solutions of differential equations 249
6.8 Local Lp second derivative estimates near the boundary 251
6.9 Existence of strong solutions for the oblique derivative problem 258
7 Viscosity Solutions of Oblique Derivative Problems 265
7.1 Definitions and notation 265
7.2 The Theorem of Aleksandrov 266
7.3 Preliminary results for the comparison theorem for viscosity solutions 274
7.4 The comparison principle for viscosity sub- and supersolutions 283
7.5 A test function construction for the oblique derivative problem 283
7.6 The comparison principle for oblique derivative problems 291
7.7 Existence and uniqueness of viscosity solutions 295
8 Pointwise Bounds for Solutions of Problems with Quasilinear Equations 301
8.1 Maximum estimates for nondivergence equations 301
8.2 Hölder estimates for nondivergence equations 305
8.3 Maximum estimates for conormal problems 307
8.4 Hölder estimates for conormal problems 313
9 Gradient Estimates for General Form Oblique Derivative Problems 321
9.1 Interior gradient bounds 321
9.2 A simple boundary value problem 329
9.3 Gradient estimates for general boundary conditions. General considerations 330
9.4 Global gradient estimates for general boundary conditions and false mean curvature equations I 333
9.5 Global gradient estimates for general boundary conditions and false mean curvature equations II 340
9.6 Local gradient estimates 346
9.7 Gradient estimates for capillary-type problems 352
10 Gradient Estimates for the Conormal Derivative Problems 365
10.1 The Sobolev inequality of Michael and Simon 365
10.2 The interior gradient bound 368
10.3 Preliminaries for estimates 382
10.4 Gradient bounds for the conormal problem 392
11 Higher Order Estimates and Existence of Solutions for Quasilinear Oblique Derivative Problems 407
11.1 The Hölder gradient estimate for conormal problems 407
11.2 A solvability theorem 410
11.3 Existence results and estimates for linear equations and nonlinear boundary conditions, in spherical caps 412
11.4 Estimates and existence results for linear equations and nonlinear boundary conditions in general domains 424
11.5 Mixed boundary value problems for simple quasilinear differential equations and nonlinear boundary conditions in spherical caps 426
11.6 Hölder gradient estimates for quasilinear equations 437
11.7 A basic existence theorem for quasilinear elliptic equations with nonlinear boundary conditions 443
11.8 Second derivative Hölder estimates 444
11.9 Existence theorems for our examples 447
12 Oblique Derivative Problems for Fully Nonlinear Elliptic Equations 457
12.1 Maximum estimates, comparison principles, and a uniqueness theorem 459
12.2 Second derivative Hölder estimates 462
12.3 Second derivative Hölder estimates for solutions of oblique derivative problems 471
12.4 Uniformly elliptic fully nonlinear problems 482
Bibliography 493
Index 507