Two-phase quadrature domains

Gardiner, Stephen J.; Sjödin, Tomas

doi:10.1007/s11854-012-0009-3

Cited by 19 publications

(17 citation statements)

References 13 publications

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“…The union of these distributions µ 1 (t), µ 2 (t) with disjoint supports (see formula (1.5)) and integrable densities, which allows a smooth evolution of the interface, is called below a twophase mother body. The notion of a mother body comes from the potential theory [16], [17], [26]- [28]. The supports of these distributions consist of sets of arcs and/or points and do not bound any two-dimensional subdomains in Ω j (t), j = 1, 2.…”

Section: A Two-phase Mother Bodymentioning

confidence: 99%

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Exact solutions to a Muskat problem with line distributions of sinks and sources

Akinyemi¹,

Savina²,

Nepomnyashchy³

2017

Contemporary Mathematics

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The evolution of a two-phase Hele-Shaw problem, a Muskat problem, under assumption of a negligible surface tension is considered. We use the Schwarz function approach and allow the sinks and sources to be line distributions with disjoint supports located in the exterior and the interior domains, a two-phase mother body. We give examples of exact solutions when the interface belongs to a certain family of algebraic curves, defined by the initial shape of the boundary, and the cusp formation does not occur.2010 Mathematics Subject Classification. Primary 76D27; Secondary 31A25, 35J65.

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Section: A Two-phase Mother Bodymentioning

confidence: 99%

“…It is also worth mentioning the recent development in the two-phase quadrature domain theory [16], [17]. In the spirit of the latter theory the problem in question could be reformulated as follows.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions to a Muskat problem with line distributions of sinks and sources

Akinyemi¹,

Savina²,

Nepomnyashchy³

2017

Contemporary Mathematics

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“…A δ-subharmonic function s = s 1 − s 2 will be undefined on the polar set Z where s 1 = −∞ = s 2 . However, as noted in [8], s has a fine limit |µ s |-almost everywhere, as well as being finely continuous at all points of Z c . We assign s this limiting value wherever it exists and, with this convention, reformulate a result of Brezis and Ponce [4] as follows.…”

Section: Tools From Potential Theory and Partial Balayagementioning

confidence: 89%

“…We assign s this limiting value wherever it exists and, with this convention, reformulate a result of Brezis and Ponce [4] as follows. A short proof of it may be found in [8].…”

Section: Tools From Potential Theory and Partial Balayagementioning

confidence: 95%

Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions

Gardiner

Sjödin

2014

Arch Rational Mech Anal

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It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.

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“…As far as we know, the Sakai's concentration condition together with estimates of the one phase solutions of µ ± is a sufficient condition (see [7]). For more existence result see the recent article [8].…”

Section: Discussion On Existence Theorymentioning

confidence: 99%

Some properties of two-phase quadrature domains

Babaoglu

Bazarganzadeh

2011

Nonlinear Analysis: Theory, Methods & Applications

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This is a submitted version of a paper published in Nonlinear Analysis.Citation for the published paper: Babaoglu, C., Bazarganzadeh, M. (2011 Abstract. In this paper, we investigate general properties of the two-phase quadrature domains, which recently has been introduced by EmamizadehPrajapat-Shahgholian. The concept, which is the generalization of the wellknown one-phase case, introduces substantial difficulties with interesting and even richer features than its one-phase counterpart. For given positive constants λ ± and two bounded and compactly supported measures µ ± , we investigate the uniqueness of the solution of the following free boundary problemwhere Ω = Ω + ∪ Ω − . It is further required that the supports of µ ± should be inside Ω ± ; this in general may fail and give rise to non-existence of solutions. Along the lines of 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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Two-phase quadrature domains

Cited by 19 publications

References 13 publications

Exact solutions to a Muskat problem with line distributions of sinks and sources

Exact solutions to a Muskat problem with line distributions of sinks and sources

Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions

Some properties of two-phase quadrature domains

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