A Geometric Approach to Free Boundary Problems Luis Caffarelli Sandro Salsa Graduate Studies in Mathematics Volume 68 a,,,,. n American Mathematical Society Providence, Rhode Island
Introduction vii Part 1. Elliptic Problems 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 C 1 ' 7 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 i?-subsolutions 53 4.5. Free boundary improvement. Basic iteration 62 iii
iv a, 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 e-monotone functions 72 5.4. A continuous family of i?-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. wis 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 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. i?-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 u v at a "contact point" 172 10.4. A continuous family of perturbations 174 10.5. Improvement of e-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 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. e-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(l, 1/2) domains 235 13.2. Caloric functions in Lipschitz domains 241 13.3. Asymptotic behavior near the zero set 248 13.4. e-monotonicity and full monotonicity 256
vi 13.5. An excursion on caloric measure 262 Bibliography 265 Index 269