自由边界问题的几何方法 影印版PDF电子书下载

查看更多关于的内容

Part 1.Elliptic Problems 3

Chapter 1.An Introductory Problem 3

1.1.Introduction and heuristic considerations 3

1.2.A one-phase singular perturbation problem 6

1.3.The free boundary condition 17

Chapter 2.Viscosity Solutions and Their Asymptotic Developments 25

2.1.The notion of viscosity solution 25

2.2.Asymptotic developments 27

2.3.Comparison principles 30

Chapter 3.The Regularity of the Free Boundary 35

3.1.Weak results 35

3.2.Weak results for one-phase problems 36

3.3.Strong results 40

Chapter 4.Lipschitz Free Boundaries Are C1，γ 43

4.1.The main theorem.Heuristic considerations and strategy 43

4.2.Interior improvement of the Lipschitz constant 47

4.3.A Harnack principle.Improved interior gain 51

4.4.A continuous family of R-subsolutions 53

4.5.Free boundary improvement.Basic iteration 62

Chapter 5.Flat Free Boundaries Are Lipschitz 65

5.1.Heuristic considerations 65

5.2.An auxiliary family of functions 70

5.3.Level surfaces of normal perturbations of ε-monotone functions 72

5.4.A continuous family of R-subsolutions 74

5.5.Proof of Theorem 5.1 76

5.6.A degenerate case 80

Chapter 6.Existence Theory 87

6.1.Introduction 87

6.2.u＋is locally Lipschitz 90

6.3.u is Lipschitz 91

6.4.u＋is nondegenerate 95

6.5.u is a viscosity supersolution 96

6.6.u is a viscosity subsolution 99

6.7.Measure-theoretic properties of F（u） 101

6.8.Asymptotic developments 103

6.9.Regularity and compactness 106

Part 2.Evolution Problems 111

Chapter 7.Parabolic Free Boundary Problems 111

7.1.Introduction 111

7.2.A class of free boundary problems and their viscosity solutions 113

7.3.Asymptotic behavior and free boundary relation 116

7.4.R-subsolutions and a comparison principle 118

Chapter 8.Lipschitz Free Boundaries： Weak Results 121

8.1.Lipschitz continuity of viscosity solutions 121

8.2.Asymptotic behavior and free boundary relation 124

8.3.Counterexamples 125

Chapter 9.Lipschitz Free Boundaries： Strong Results 131

9.1.Nondegenerate problems： main result and strategy 131

9.2.Interior gain in space （parabolic homogeneity） 135

9.3.Common gain 138

9.4.Interior gain in space （hyperbolic homogeneity） 141

9.5.Interior gain in time 143

9.6.A continuous family of subcaloric functions 149

9.7.Free boundary improvement.Propagation lemma 153

9.8.Regularization of the free boundary in space 157

9.9.Free boundary regularity in space and time 160

Chapter 10.Flat Free Boundaries Are Smooth 165

10.1.Main result and strategy 165

10.2.Interior enlargement of the monotonicity cone 168

10.3.Control of uv at a “contact point” 172

10.4.A continuous family of perturbations 174

10.5.Improvement of ε-monotonicity 177

10.6.Propagation of cone enlargement to the free boundary 180

10.7.Proof of the main theorem 183

10.8.Finite time regularization 185

Part 3.Complementary Chapters： Main Tools 191

Chapter 11.Boundary Behavior of Harmonic Functions 191

11.1.Harmonic functions in Lipschitz domains 191

11.2.Boundary Harnack principles 195

11.3.An excursion on harmonic measure 201

11.4.Monotonicity properties 203

11.5.ε-monotonicity and full monotonicity 205

11.6.Linear behavior at regular boundary points 207

Chapter 12.Monotonicity Formulas and Applications 211

12.1.A 2-dimensional formula 211

12.2.The n-dimensional formula 214

12.3.Consequences and applications 222

12.4.A parabolic monotonicity formula 230

12.5.A singular perturbation parabolic problem 233

Chapter 13.Boundary Behavior of Caloric Functions 235

13.1.Caloric functions in Lip（1， 1/2） domains 235

13.2.Caloric functions in Lipschitz domains 241

13.3.Asymptotic behavior near the zero set 248

13.4.ε-monotonicity and full monotonicity 256

13.5.An excursion on caloric measure 262

Bibliography 265

Index 269

查看更多关于的内容