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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
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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

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