Regularity of a free boundary with application to the Pompeiu problem

Annals of Mathematics

Volumn 151, Issue 1, 2000, Pages 269-292

  Caffarelli, Luis A ^a   Karp, Lavi ^b   Shahgholian, Henrik ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b International College of Technology   (Israel)

^c ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

Free boundary problems; Pompeiu problem; Regularity

Indexed keywords

EID: 0034344817     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/121117     Document Type: Article

References (30)

1. H. W. ALT, L. A. CAFFARELLI, and A. FRIEDMAN, Variational problems with two phases and their free boundaries, Trans. A.M.S. 282 (1984), 431-461.
2. I. ATHANASOPOULOS and L. A. CAFFARELLI, A theorem of real analysis and its application to free boundary problems, Comm. Pure Appl. Math. 38 (1985), 499-502.
3. W. BECKNER, C. KENIG, and J. PIPHER, in preparation.
4. L. A. CAFFARELLI, The regularity of free boundaries in higher dimension, Acta Math. 139 (1977), 155-184.
5. _, Compactness methods in free boundary problems, Comm. P.D.E. 5 (1980), 427-448.
6. _, A Harnack inequality approach to the regularity of free boundaries, Scoula Norm. Sup. Pisa 15 (1988), 583-602.
7. _, The obstacle problem revisited, III. Existence theory, compactness and dependence on X, J. Fourier Anal. Appl. 4 (1998), 383-402.
8. L. A. CAFFARELLI and C. KENIG, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. of Math. 120 (1998), 391-440.
9. E. DI BENEDETTO and A. FRIEDMAN, Bubble growth in porous media, Indiana Univ. Math. J. 35 (1986), 573-606.
10. S. FRIEDLAND and W. K. HAYMAN, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), 133-161.
11. nd edition, Springer-Verlag, New York, 1983.
12. B. GUTAFSSON and M. PUTINAR, The exponential transform and regularity of free boundaries in two dimensions, Ann. Scoula Norm. Sup. Pisa Cl. Sci. 26 (1998), 507-543.
13. V. ISAKOV, Inverse Source Problems, A.M.S. Math. Surveys and Monographs 34, Providence, RI, 1990.
14. _, Inverse theorems on the smoothness of potentials, in Differential Equations 11 (1976), 50-57 (translated from Russian).
15. L. KARP and H. SHAHGHOLIAN, Regularity of a free boundary problem, J. Geometric Analysis, to appear.
16. _, Regularity of a free boundary at the infinity point, Comm. Partial Differential Eq., to appear.
17. D. KINDERLEHRER and L. NIRENBERG, Regularity in free boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 373-391.
18. p-densities and its applications to inverse problems of gravimetry, in Theory and Practice of Gravitational and Magnetic Fields Interpretation in USSR, 188-197, Naukova Dumka Press, Kiev, 1983 (Russian).
19. J. MCCARTHY and L. YANG, Subnormal operators and quadrature domains. Adv. Math. 127 (1997), 52-72.
20. M. PUTINAR, Extremal solutions of the two dimensional L-problems of moments, J. Funct. Anal. 136 (1996), 331-364.
21. M. SAKAI. Quadrature Domains, Lecture Notes in Math. 934, Springer-Verlag, New York, 1982.
22. _, Regularity of a boundary having a Schwarz function, Acta Math. 166 (1991). 263-297.
23. _, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 (1993), 341-364.
24. H. Shahgholian, On quadrature domains and the Schwarz potential, J. Math. Anal. 171 (1992), 61-78.
25. H. S. SHAPIRO, Global geometric aspects of Cauchy's problem for the Laplace operator, in Geometrical and Algebraical Aspects in Several Complex Variables, 309-324 , Cetraro, 1989.
26. _, The Schwarz Function and Its Generalization to Higher Dimensions, Univ. of Arkansas Lecture Notes in the Math. Sci. 9. John Wiley & Sons, Inc., New York, 1992.
27. E. SPERNER, Zur symmetrisierung von funktionen auf Sphären, Math. Z. 134 (1973), 317-327.
28. J. SPRUCK, Uniqueness in diffusion model of population biology, Comm. P.D.E. 8 (1983), 1605-1620.
29. V. N. STRAKHOV, The inverse logarithmic potential problem for contact surface, Physics of the Solid Earth 10 (1974), 104-114 (translated from Russian).
30. S. WILLIAMS, Analyticity of the boundary for Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J. 30 (1981), 357-369.