Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T01:55:23.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear combinations of harmonic measures and quadrature domains of signed measures with small supports

Published online by Cambridge University Press:  20 January 2009

Makoto Sakai
Makoto Sakai
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-Shi, Tokyo, 192-0397, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we discuss the shape of the quadrature domain of a signed measure for harmonic functions. It is known that the quadrature domain of a positive measure with small support is like a ball if the total measure is large enough. We show that, on the contrary, if the measure is not positive then the quadrature domain can be close to an arbitrary domain. This follows from a lemma concerning linear combinations of harmonic measures.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 433 - 444
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Aikawa, H., Sets of determination for harmonic functions in an NTA domain, J Math. Soc. Japan 48 (1996), 299–315.CrossRefGoogle Scholar
2.Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory (Graduate Texts in Mathematics 137, Springer-Verlag, Berlin, 1992).CrossRefGoogle Scholar
3.Bass, R. F. and Burdzy, K., The Martin boundary in non-Lipschitz domains, Trans. Amer. Math. Soc. 337 (1993), 361–378.CrossRefGoogle Scholar
4.Bonsall, F. F., Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc. (2) 30 (1987), 471–477.CrossRefGoogle Scholar
5.Bonsall, F. F. and Walsh, D., Vanishing ℓ1-sums of the Poisson kernel, and sums with positive coefficients, Proc. Edinburgh Math. Soc. (2) 32 (1989), 431–447.CrossRefGoogle Scholar
6.Constantinescu, C. and Cornea, A., Ideale Ränder Riemannsher Flächen (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
7.Constantinescu, C. and Cornea, A., Potential Theory on Harmonic Spaces (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
8.Ward, N. F. Dudley, On a decomposition theorem for continuous functions of Hayman and Lyons, New Zealand J. Math. 22 (1993), 49–59.Google Scholar
9.Essén, M., On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc. (2) 36 (1993), 87–106.CrossRefGoogle Scholar
10.Gardiner, S. J., Sets of determination for harmonic functions, Trans. Amer. Math. Soc. 338 (1993), 233–243.CrossRefGoogle Scholar
11.Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed (Springer-Verlag, Berlin, 1983).Google Scholar
12.Gustafsson, B., A distortion theorem for quadrature domains for harmonic functions, J Math. Anal. Appl. 202 (1996), 169–182.CrossRefGoogle Scholar
13.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.CrossRefGoogle Scholar
14.Hayman, W. K., Atomic decompositions, in Recent Advances in Fourier Analysis and its Applications (Byrnes, J. S. and Byrnes, J. L., eds., Kluwer Academic Publ., Dordrecht, 1990), 597–611.CrossRefGoogle Scholar
15.Hayman, W. K. and Lyons, T. J., Bases for positive continuous functions, J. London Math. Soc. (2) 42 (1990), 292–308.CrossRefGoogle Scholar
16.Helms, L. L., Introduction to Potential Theory (John Wiley and Sons, New York, 1969).Google Scholar
17.Hunt, R. A. and Wheeden, R. L., Positive harmonic functions on Lipshitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527.CrossRefGoogle Scholar
18.Jerison, D. S. and Kenig, C. E., Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math. 46 (1982), 80–147.CrossRefGoogle Scholar
19.Pommerenke, C., Boundary Behaviour of Conformal Maps (Springer-Verlag, Berlin, 1992).CrossRefGoogle Scholar
20.Sakai, M., Quadrature domains (Lecture Notes in Math. 934, Springer-Verlag, Berlin, 1982).CrossRefGoogle Scholar
21.Sakai, M., Applications of variational inequalities to the existence theorem on quadrature domains, Trans. Amer. Math. Soc. 276 (1983), 267–279.CrossRefGoogle Scholar
22.Sakai, M., Solutions to the obstacle problem as Green potentials, J. Anal. Math. 44 (1985), 97–116.CrossRefGoogle Scholar
23.Sakai, M., Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow, Potential Anal. 8 (1998), 277–302.CrossRefGoogle Scholar
24.Widman, K.-O., Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37.CrossRefGoogle Scholar