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Some properties of two-phase quadrature domains

โœ Scribed by Ceni Babaoglu; Mahmoudreza Bazarganzadeh

Publisher
Elsevier Science
Year
2011
Tongue
English
Weight
250 KB
Volume
74
Category
Article
ISSN
0362-546X

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โœฆ Synopsis

In this paper, we investigate general properties of the two-phase quadrature domain, which was recently introduced by Emamizadeh, Prajapat and Shahgholian. The concept, which is a generalization of the well-known one-phase domain, introduces substantial difficulties with interesting features even richer than those of the one-phase counterpart.

For given positive constants ฮป ยฑ and two bounded and compactly supported measures ยต ยฑ , we investigate the uniqueness of the solution of the following free boundary problem:

where โ„ฆ = โ„ฆ + โˆช โ„ฆ -. It is further required that the supports of ยต ยฑ should be inside โ„ฆ ยฑ ; this in general may fail and give rise to non-existence of solutions.

Along the paths to various properties that we state and prove here, we also present several conjectures and open problems that we believe should be true.

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