scholarly journals Quasi-nearly subharmonic functions and conformal mappings

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Vesna Kojic
Keyword(s):  
Conformal Mapping ◽  
Subharmonic Function ◽  
Conformal Mappings ◽  
Subharmonic Functions ◽  
Similar Fact ◽  
Oscillating Functions

If is a conformal mapping and u is a quasi-nearly subharmonic function, then u o ? is quasi-nearly subharmonic. A similar fact for "regularly oscillating" functions holds. .

scholarly journals Bi-Lipschitz Mappings and Quasinearly Subharmonic Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Oleksiy Dovgoshey ◽  
Juhani Riihentaus
Keyword(s):  
Mean Value ◽  
Conformal Mappings ◽  
Plane Case ◽  
Subharmonic Functions ◽  
Invariance Properties ◽  
Lipschitz Mappings ◽  
Mean Value Inequality ◽  
Generalized Mean ◽  
Oscillating Functions

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalizations to her results by showing that inℝn,n≥2, these both classes are invariant under bi-Lipschitz mappings.

scholarly journals Conformal mapping of the Misner–Sharp mass from gravitational collapse

2016 ◽  
Vol 25 (07) ◽  
pp. 1650081 ◽  
Author(s):  
Fayçal Hammad
Keyword(s):  
Conformal Mapping ◽  
Gravitational Collapse ◽  
Conformal Transformation ◽  
Conformal Mappings ◽  
Conformal Transformations ◽  
Theories Of Gravity ◽  
Definition Of ◽  
Geometric Definition

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

scholarly journals Dirichlet Problems on Riemann Surfaces and Conformal Mappings

1951 ◽  
Vol 3 ◽  
pp. 91-137 ◽  
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Conformal Mappings ◽  
Dirichlet Problems ◽  
Boundary Correspondence ◽  
Abstract Riemann Surface

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

scholarly journals Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains

2020 ◽  
Author(s):  
Haiqing Xu
Keyword(s):  
Conformal Mapping ◽  
Unit Disk ◽  
Conformal Mappings ◽  
Optimal Regularity ◽  
Finite Distortion ◽  
Homeomorphic Extension

AbstractThe conformal mapping $$f(z)=(z+1)^2 $$f(z)=(z+1)2 from $${\mathbb {D}}$$D onto the standard cardioid has a homeomorphic extension of finite distortion to entire $${\mathbb {R}}^2 .$$R2. We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $${\mathbb {D}}$$D onto cardioid-type domains.

On conformal mappings of domains of infinite connectivity

1955 ◽  
Vol 51 (1) ◽  
pp. 56-64 ◽  
Author(s):  
L. J. M. Brown
Keyword(s):  
Conformal Mapping ◽  
Complex Plane ◽  
Conformal Mappings

1. In the theory of conformal mapping, there are many problems associated with domains of infinite connectivity which have not yet been fully explored. We consider, in this paper, some of these problems in relation to domains in the closed complex plane. In the following all domains are assumed to be plane domains and all mappings to be (1, 1) and conformal in the domains in which they are defined.

Montel's theorem for subharmonic functions and solutions of partial differential equations

1971 ◽  
Vol 69 (1) ◽  
pp. 123-150 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Compact Subset ◽  
Subharmonic Function ◽  
Partial Differential ◽  
Subharmonic Functions ◽  
Half Strip ◽  
Montel’S Theorem ◽  
Image Position ◽  
Suitable Sequence

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-stripand if there exists an x0 in (a, b) such thatas y → ∞, thenas y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and ifas y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such thatas y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)asy → ∞, thenas y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker conditionas n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.

scholarly journals Approximate Conformal Mappings and Elasticity Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Pyotr N. Ivanshin ◽  
Elena A. Shirokova
Keyword(s):  
Conformal Mapping ◽  
Elasticity Theory ◽  
Unit Disk ◽  
Connected Domain ◽  
Smooth Boundary ◽  
Mapping Function ◽  
Conformal Mappings ◽  
New Method ◽  
Taylor Polynomial ◽  
Elasticity Problems

Here, we present the new method of approximate conformal mapping of the unit disk to a one-connected domain with smooth boundary without auxiliary constructions and iterations. The mapping function is a Taylor polynomial. The method is applicable to elasticity problems solution.

scholarly journals Asymptotics of $\delta$-subharmonic functions of finite order

2020 ◽  
Vol 54 (2) ◽  
pp. 188-192
Author(s):  
M.V. Zabolotskyi
Keyword(s):  
Finite Order ◽  
Subharmonic Function ◽  
Entire Functions ◽  
Proximate Order ◽  
Subharmonic Functions ◽  
Riesz Measure

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positiveorder we found the asymptotical representation of the form\[u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,\]where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.

A Study on Optimum Topology Using Conformal Mapping

2000 ◽  
Author(s):  
Satoshi Kitayama ◽  
Hiroshi Yamakawa
Keyword(s):  
Conformal Mapping ◽  
Fluid Mechanics ◽  
Laplace Equation ◽  
Coordinate Transformation ◽  
Theory Of Elasticity ◽  
Conformal Mappings ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Planar Structures ◽  
Optimum Topology

Abstract This paper presents a method to determine optimum topologies of two dimensional elastic planar structures by using conformal mappings. We use the conformal mappings which is known to be effective in two dimensional fluid mechanics, electromagnetics and elasticity by complex coordinate transformation. We show that two invariants of stress can satisfy the Laplace equation, and then we clarify that corresponding relationships between fluid mechanics and electromagnetics can also be valid in the theory of elasticity. Then, presented a method to obtain optimum topologies is easier than by the conventional methods. We treated several numerical examples by the presented method. Through numerical examples, we can examine the effectiveness of the proposed method.

scholarly journals Experiment on numerical conformal mapping of unbounded multiply connected domain in fundamental solutions method

2001 ◽  
Vol 26 (1) ◽  
pp. 55-63
Author(s):  
Tetsuo Inoue ◽  
Hideo Kuhara ◽  
Kaname Amano ◽  
Dai Okano
Keyword(s):  
Conformal Mapping ◽  
Fundamental Solutions ◽  
High Accuracy ◽  
Conformal Mappings ◽  
Multiply Connected Domains ◽  
Multiply Connected Domain ◽  
Numerical Conformal Mapping ◽  
Multiply Connected ◽  
Weighted Polynomials ◽  
Numerical Conformal Mappings

We are concerned with the experiment on numerical conformal mappings. A potentially theoretical scheme in the fundamental solutions method, different from the conventional one, has been recently proposed for numerical conformal mappings of unbounded multiply connected domains. The scheme is based on the asymptotic theorem on extremal weighted polynomials. The scheme has the characteristic called “invariant and dual.” Applying the scheme for typical examples, we will show that the numerical results of high accuracy may be obtained.

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