A Geometric Approach to Free Boundary Problems
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1 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
2 Introduction vii Part 1. Elliptic Problems Chapter 1. An Introductory Problem Introduction and heuristic considerations A one-phase singular perturbation problem The free boundary condition 17 Chapter 2. Viscosity Solutions and Their Asymptotic Developments The notion of viscosity solution Asymptotic developments Comparison principles 30 Chapter 3. The Regularity of the Free Boundary Weak results Weak results for one-phase problems Strong results 40 Chapter 4. Lipschitz Free Boundaries Are C 1 ' The main theorem. Heuristic considerations and strategy Interior improvement of the Lipschitz constant A Harnack principle. Improved interior gain A continuous family of i?-subsolutions Free boundary improvement. Basic iteration 62 iii
3 iv a, Chapter 5. Flat Free Boundaries Are Lipschitz Heuristic considerations An auxiliary family of functions Level surfaces of normal perturbations of e-monotone functions A continuous family of i?-subsolutions Proof of Theorem A degenerate case 80 Chapter 6. Existence Theory Introduction u + is locally Lipschitz u is Lipschitz u + is nondegenerate wis a viscosity supersolution u is a viscosity subsolution Measure-theoretic properties of F(u) Asymptotic developments Regularity and compactness 106 Part 2. Evolution Problems Chapter 7. Parabolic Free Boundary Problems Introduction A class of free boundary problems and their viscosity solutions Asymptotic behavior and free boundary relation i?-subsolutions and a comparison principle 118 Chapter 8. Lipschitz Free Boundaries: Weak Results Lipschitz continuity of viscosity solutions Asymptotic behavior and free boundary relation Counterexamples 125 Chapter 9. Lipschitz Free Boundaries: Strong Results Nondegenerate problems: main result and strategy Interior gain in space (parabolic homogeneity) Common gain Interior gain in space (hyperbolic homogeneity) 141
4 9.5. Interior gain in time A continuous family of subcaloric functions Free boundary improvement. Propagation lemma Regularization of the free boundary in space Free boundary regularity in space and time 160 Chapter 10. Flat Free Boundaries Are Smooth Main result and strategy Interior enlargement of the monotonicity cone Control of u v at a "contact point" A continuous family of perturbations Improvement of e-monotonicity Propagation of cone enlargement to the free boundary Proof of the main theorem Finite time regularization 185 Part 3. Complementary Chapters: Main Tools Chapter 11. Boundary Behavior of Harmonic Functions Harmonic functions in Lipschitz domains Boundary Harnack principles An excursion on harmonic measure Monotonicity properties e-monotonicity and full monotonicity Linear behavior at regular boundary points 207 Chapter 12. Monotonicity Formulas and Applications A 2-dimensional formula The n-dimensional formula Consequences and applications A parabolic monotonicity formula A singular perturbation parabolic problem 233 Chapter 13. Boundary Behavior of Caloric Functions Caloric functions in Lip(l, 1/2) domains Caloric functions in Lipschitz domains Asymptotic behavior near the zero set e-monotonicity and full monotonicity 256
5 vi An excursion on caloric measure 262 Bibliography 265 Index 269
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