On Domains in Which Harmonic Functions Satisfy Generalized Mean Value Properties

Potential Analysis

Volumn 7, Issue 1, 1997, Pages 467-484

  Gustafsson, Björn ^a   Sakai, Makoto ^b   Shapiro, Harold S ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b TOKYO METROPOLITAN UNIVERSITY   (Japan)

Author keywords

Balayage; Harmonic function; Integrability of harmonic functions; Martin boundary; Mean value property; Quadrature domain

Indexed keywords

EID: 0031512217     PISSN: 09262601     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1017915828219     Document Type: Article

References (28)

1. Aharonov, D. and Shapiro, H. S.: 'Domains on which analytic functions satisfy quadrature identities', J. Anal. Math. 30 (1976), 39-73.
2. Aikawa, H.: Integrability of superharmonic functions and subharmonic functions', Proc. Amer. Math. Soc. 120 (1994), 109-117.
3. Davis, P.: 'The Schwarz Function and Its Applications', Carus Math. Monographs 17, Math. Assoc. of America, 1974.
4. Doob, J. L.: Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York, 1983.
5. Dunford, N. and Schwarz, J. T.: Linear operators, volume I, Interscience, New York, 1958.
6. Gustafsson, B.: 'Existence of weak backward solutions to a generalized Hele Shaw flow moving boundary problem', Nonlinear Anal. 9 (1985), 203-215.
7. Gustafsson, B.: 'On quadrature domains and an inverse problem in potential theory', J. Anal. Math. 55 (1990), 172-216.
8. Gustafsson, B.: A distortion theorem for quadrature domains for harmonic functions, J. Math. Anal. Appl. 202 (1996), 169-182.
9. Gustafsson, B. and Sakai, M.: 'Properties of some balayage operators, with applications to quadrature domains and moving boundary problems', Nonlinear Anal. 22 (1994), 1221-1245.
10. Heins, M.: 'A lemma on positive harmonic functions', Ann. Math. 52 (1950), 568-573.
11. Helms, L. L.: Introduction to Potential Theory, John Wiley and Sons, New York, 1969.
12. Hile, G. N. and Yeh, R. Z.: 'Phragmén-Lindelöf principles for solutions of elliptic differential inequalities', J. Math. Anal. Appl. 107 (1985), 478-497.
13. Maeda, F. Y. and Suzuki, N.: 'The integrability of superharmonic functions on Lipschitz domains', Bull. London Math. Soc. 21 (1989), 270-278.
14. Margulis, A. S.: The moving boundary problem of potential theory, Adv. Math. Sci. Appl. 5 (1995).
15. Margulis, A. S.: On the Inverse Problem of Potential Theory, in press.
16. Miller, K.: 'Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications', Ann. Mat. Pura Appl. 90 (1971), 297-329.
17. Randol, B.: 'Some remarks on the stability of a property related to the mean value theorem for harmonic functions', Proc. Amer. Math. Soc. 114 (1992), 175-179.
18. Sakai, M.: 'Quardrature Domains', Lecture Notes in Math. 934, Springer-Verlag, Berlin, 1982.
19. Sakai, M.: 'Regularity of a boundary having a Schwarz function', Acta Math. 166 (1991), 263-297.
20. Sakai, M.: 'Regularity of boundaries of quadrature domains in two dimensions', SIAM J. Math. Anal. 24 (1993), 341-364.
21. Sakai, M.: 'Regularity of free boundaries in two dimensions', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 323-339.
22. Sakai, M.: Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow, preprint 1995.
23. Schaeffer, D. G.: 'Some examples of singularities in a free boundary', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 131-144.
24. Shapiro, H. S.: The Schwarz Function and Its Generalization to Higher Dimensions, Wiley and Sons, New York, 1992.
25. Shields, A. L.: 'Weighted shift operators and analytic function theory', Math. Surveys vol. 13, Amer. Math. Soc. 1974.
26. Stout, E. L.: 'Harmonic duality, hyperfunctions and removable singularities', Izvestiya. Ross. Akad. Nauk. Ser. Mat. (in press).
27. Suzuki, N.: 'Note on the integrability of superharmonic functions', Proc. Amer. Math. Soc 118 (1993), 415-417.
28. Zorn, P.: 'Analytic functionals and Bergman spaces', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), 365-404.