Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

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Integral transforms for logharmonic mappings

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02578-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hugo Arbeláez ◽

Víctor Bravo ◽

Rodrigo Hernández ◽

Willy Sierra ◽

Osvaldo Venegas

Keyword(s):

Function Theory ◽

Classical Problem ◽

Integral Transforms ◽

Geometric Function Theory ◽

Integral Transformations ◽

Geometric Function

AbstractBieberbach’s conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type $f_{\alpha }(z)=\int _{0}^{z}(f(\zeta )/\zeta )^{\alpha }\,d\zeta $ f α ( z ) = ∫ 0 z ( f ( ζ ) / ζ ) α d ζ or $F_{\alpha }(z)=\int _{0}^{z}(f'(\zeta ))^{\alpha }\,d\zeta $ F α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ appear. In this note we extend the classical problem of finding the values of $\alpha \in \mathbb{C}$ α ∈ C for which either $f_{\alpha }$ f α or $F_{\alpha }$ F α are univalent, whenever f belongs to some subclasses of univalent mappings in $\mathbb{D}$ D , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario.

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Some estimates for extremal decomposition of the complex plane

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-5 ◽

2018 ◽

Vol 32 ◽

pp. 42-47

Author(s):

Iryna Denega

Keyword(s):

Complex Variable ◽

Open Problem ◽

Function Theory ◽

Extremal Problems ◽

Geometric Function Theory ◽

Geometric Function ◽

Inner Radius ◽

Overlapping Domains ◽

Disjoint Domains

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. In this paper, we consider the well-known problem of maximum of the functional $r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)$, where $B_{0}$,..., $B_{n}$ are pairwise disjoint domains in $\overline{\mathbb{C}}$, $ a_0=0 $, $|a_{k}|=1$, $k=\overline{1,n}$ are different points of the circle, $\gamma\in (0, n]$, and $r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ relative to the point $ a $. This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on $\overline{\mathbb{C}}$.

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