This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

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-Bakel0man-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 H¨older 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 H¨older spaces 47

2.2 An alternative characterization of H¨older 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. H¨older 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 H¨older 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 Poincar´e’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 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 H¨older 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 H¨older estimates for nondivergence equations 305

8.3 Maximum estimates for conormal problems 307

8.4 H¨older 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 H¨older 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 H¨older 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 H¨older 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 H¨older estimates 462

12.3 Second derivative H¨older estimates for solutions of oblique derivative problems 471

12.4 Uniformly elliptic fully nonlinear problems 482

Bibliography 493

Index 507

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