Mother bodies of algebraic domains in the complex plane

Sjödin, Tomas

doi:10.1080/17476930600610049

Cited by 14 publications

(4 citation statements)

References 4 publications

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“…Conditions (M1)-(M4) are, generally speaking, quite demanding, and consequently domains with mother bodies are somewhat rare. Cases where the existence of the mother body is known include discs, convex polyhedra [25] and ellipses [49], and mother bodies have also been numerically obtained for certain oval shapes [47]. Mother body measures appear in the context of quadrature domains [26], inverse problems in geophysics [54], zero distribution of orthogonal polynomials [28,43], among others.…”

Section: 2mentioning

confidence: 99%

The mother body phase transition in the normal matrix model

Bleher¹,

Silva²

2016

Preprint

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The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth.In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω that we determine explicitly by finding the rational parametrization of its boundary.We also study in details the mother body problem associated to Ω. It turns out that the mother body measure µ * displays a novel phase transition that we call the mother body phase transition: although ∂Ω evolves analytically, the mother body measure undergoes a "one-cut to three-cut" phase transition.To construct the mother body measure, we define a quadratic differential on the associated spectral curve, and embed µ * into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on µ * . In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly.Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated g-functions significantly more involved, and the critical graph of becomes the key technical tool in this analysis as well.

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Section: 2mentioning

confidence: 99%

The mother body phase transition in the normal matrix model

Bleher¹,

Silva²

2016

Preprint

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“…1) Moment problems, Operator Theory (hyponormal operators), Exponential Transforms (see [12,13,20,21,22,23,24,37,38,39], 2) Hele-Shaw related problems (see [5,17,18]), 3) Problems related to Bergman and Szegö Kernels (see [3,4]), 4) Mother bodies and skeletons (see [11,15,28,36], 5) The Cauchy Problem in C n (see [7,19,34]), 6) Quadrature Surfaces (see [30,31] . It is tantalizing and wishful to think that many of these concepts can be developed also in the two-phase situation.…”

Section: Further Perspectivesmentioning

confidence: 99%

“…2) Hele-Shaw related problems (see [5,17,18]), 3) Problems related to Bergman and Szegö Kernels (see [3,4]), 4) Mother bodies and skeletons (see [11,15,28,36],…”

Section: Further Perspectivesmentioning

confidence: 99%

Harmonic balls and the two-phase Schwarz function

Shahgholian¹,

Sjödin²

2013

Complex Variables and Elliptic Equations

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In this article, we introduce the concept of harmonic balls in sub-domains of n , through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

Funding Agencies|Swedish Research Council|

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“…During the last decades the notion of a motherbody of a solid domain or, more generally, of a positive Borel measure was discussed both in geophysics and mathematics, see e.g. [32], [28], [15], [35]. It was apparently pioneered in the 1960's by a Bulgarian geophysicist D. Zidarov [35] and later mathematically developed by B. Gustafsson [15].…”

Section: Introductionmentioning

confidence: 99%

On mother body measures with algebraic Cauchy transform

Bøgvad¹,

Shapiro²

2017

Enseign. Math.

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Abstract. Below we discuss the existence of a motherbody measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in C. Several conjectures and open problems are posed.

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Mother bodies of algebraic domains in the complex plane

Cited by 14 publications

References 4 publications

The mother body phase transition in the normal matrix model

The mother body phase transition in the normal matrix model

Harmonic balls and the two-phase Schwarz function

On mother body measures with algebraic Cauchy transform

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