Free boundary regularity for harmonic measures and Poisson kernels

Annals of Mathematics

Volumn 150, Issue 2, 1999, Pages 369-454

  Kenig, Carlos E ^a   Toro, Tatiana ^b  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033235413     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/121086     Document Type: Article

References (35)

1. H. W. ALT and L. A. CAFFARELLI, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144.
2. R. COIFMAN and G. WEISS, Extensions of Hardy spaces and their use in analysis, Bull. A.M.S. 83 (1977), 569-645.
3. 1,α, Rev. Mat. Iberoamericana 3 (1987), 139-162.
4. _, A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), 55-78.
5. G. DAVID, Opérateurs d' integrale singulière sur les surfaces régulières, Ann. Scient. École Norm. Sup., Ser. 4 21 (1988), 225-258.
6. G. DAVID and D. JERISON, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), 831-845.
7. G. DAVID, C. KENIG, and T. TORO, Regularity of sets supporting asymptotically optimally doubling measures, in preparation.
8. G. DAVID and S. SEMMES, Analysis of and on Uniformly Rectifiable Sets, Math. Surveys and Mono. , 38, A.M.S. series, 1993.
9. p Spaces, Academic Press, New York, 1970.
10. L. C. EVANS and R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Adv. Math., CRC Press, Boca Raton, FL, 1992.
11. J. GARCÍA-CUERVA and J. L. RUBIO DE FRANCIA, Weighted Norm Inequalities and Related Topics, Math. Studies 116, North Holland, 1985.
12. M. GROMOV, Structures Métriques pour les Variétés Riemanniennes (J. Lafontaine & P. Pansu, eds.), Textes Math., CEDIC/Fernand-Nathan, Paris, 1981.
13. L. L. HELMS, Introduction to Potential Theory, Pure Appl. Math. XXII, Robert E. Krieger Pub. Co., Huntington, NY, 1975.
14. D. JERISON, Regularity of the Poisson kernel and free boundary problems, Colloq. Math. 60-61 (1990), 547-567.
15. D. JERISON and C. KENIG, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80-147.
16. 1 domain has vanishing mean oscillation, Trans. A.M.S. 273 (1982), 781-794.
17. _, An identity with applications to harmonic measure, Bull. A.M.S. 2 (1980), 447-451.
18. C. KENIG and T. TORO, Harmonic measure on locally flat domains, Duke Math. J. 87, (1997), 509-551.
19. _, Poisson kernel characterization of chord arc domains with small constant, in preparation.
20. O. KOWALSKI and D. PREISS, Besicovitch-type properties of measures and submanifolds, J. Reine Angew. Math. 379 (1987), 115-151.
21. P. MATTILA, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Adv. Math. 44, Cambridge University Press, 1995.
22. C. B. MORREY, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.
23. CH. POMMERENKE, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), 199-208.
24. n: Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643.
25. E. REIFENBERG, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1-92.
26. S. SEMMES, Chord-arc surfaces with small constant, I, Adv. in Math. 85 (1991), 198-223.
27. _, Chord-arc surfaces with small constant, II: Good parametrizations, Adv. in Math. 88 (1991) 170-199.
28. _, Analysis vs. geometry on a class of rectifiable hypersurfaces, Indiana Univ. Math. J. 39 (1990), 1005-1035.
29. L. SIMON, Lectures on geometric measure theory, Australian National University, 1983.
30. _, Rectifiability of the singular set of energy minimizing maps, Calc. of Variations and PDE 3 (1995), 1-65.
31. _, Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps, Surveys in Diff. Geom. 2 (1995), 246-305.
32. E. M. STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and oscillatory Integrals, Princeton Math. Series, Princeton Univ. Press, Princeton, NJ, 1993.
33. T. TORO, Geometric conditions and existence of bilipschitz parameterizations, Duke Math. J. 77 (1995), 193-227.
34. N. WIENER, The Dirichlet problem, J. Math. Phys. 3 (1924), 127-146.
35. A. ZYGMUND, Trigonometric Series, Vol. 1, Cambridge Univ. Press, 1959.