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Some properties of two-phase quadrature domains
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 10, 3386-3396 p.Article in journal (Refereed) Published
Abstract [en]

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 lambda(+/-) and two bounded and compactly supported measures mu(+/-), we investigate the uniqueness of the solution of the following free boundary problem: {Delta u =(lambda(+)chi(Omega)+ - mu(+)) - (lambda(-)chi(Omega)- - mu(-)), in R(N)(N >= 2), u = 0, in R(N)\Omega, (1) where Omega = Omega(+) boolean OR Omega(-). It is further required that the supports of mu(+/-) should be inside Omega(+/-); 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.

Place, publisher, year, edition, pages
2011. Vol. 74, no 10, 3386-3396 p.
Keyword [en]
Quadrature domain, Two-phase free boundary problems, Uniqueness
National Category
URN: urn:nbn:se:su:diva-66980DOI: 10.1016/ 000289191800026OAI: diva2:470078

authorCount :2

Available from: 2011-12-28 Created: 2011-12-22 Last updated: 2012-10-18Bibliographically approved

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