scholarly journals Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fagueye Ndiaye
Keyword(s):  
Fixed Point ◽  
Conformal Mapping ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Bounded Domain ◽  
Cauchy Data ◽  
Two Dimensions ◽  
Inverse Eigenvalue Problem ◽  
Unknown Boundary ◽  
Domain Identification

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.

scholarly journals Inner composition of analytic mappings on the unit disk

1991 ◽  
Vol 14 (2) ◽  
pp. 221-226 ◽  
Author(s):  
John Gill
Keyword(s):  
Fixed Point ◽  
Unit Disk ◽  
Analytic Functions ◽  
Continued Fraction ◽  
Unique Fixed Point ◽  
Iteration Theory ◽  
Basic Theorem ◽  
Closed Unit Disk ◽  
Compositional Structure ◽  
Compositional Structures

A basic theorem of iteration theory (Henrici [6]) states thatfanalytic on the interior of the closed unit diskDand continuous onDwithInt(D)f(D)carries any pointz ϵ Dto the unique fixed pointα ϵ Doff. That is to say,fn(z)→αasn→∞. In [3] and [5] the author generalized this result in the following way: LetFn(z):=f1∘…∘fn(z). Thenfn→funiformly onDimpliesFn(z)λ, a constant, for allz ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structuresf1∘…∘fn(z)where thefj's are analytic onInt(D)and continuous onDwithInt(D)fj(D), but essentially random. Applications include analytic functions defined by this process.

The Lindelöf principle in ℂn

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Peter Dovbush
Keyword(s):  
Holomorphic Function ◽  
Bounded Domain ◽  
Lipschitz Condition ◽  
Function Theory ◽  
Riemann Sphere ◽  
Normal Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Kobayashi Metric ◽  
Lindelöf Principle

AbstractLet D be a bounded domain in ℂn. A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.

scholarly journals Boundary behaviour of holomorphic functions on the cardioid domain with some applications

2019 ◽  
Vol 38 (7) ◽  
pp. 203-218
Author(s):  
Shatha Sami Alhily ◽  
_ Deepmala
Keyword(s):  
General Solution ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Laplace Equation ◽  
Holomorphic Functions ◽  
Research Paper ◽  
Boundary Behaviour ◽  
Polar Function

The objective of this research paper is to show how the Bennan'sconjecture  become a useful tool  to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.

scholarly journals Handel’s fixed point theorem revisited

2012 ◽  
Vol 33 (5) ◽  
pp. 1584-1610
Author(s):  
JULIANA XAVIER
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unit Disk ◽  
Point Theorem ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Disk ◽  
Oriented Cycle

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

scholarly journals DILATONIC GRAVITY NEAR TWO DIMENSIONS AS A STRING THEORY

1995 ◽  
Vol 10 (27) ◽  
pp. 2001-2008 ◽  
Author(s):  
E. ELIZALDE ◽  
S.D. ODINTSOV
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
String Theory ◽  
Sigma Model ◽  
Scalar Fields ◽  
Two Dimensions ◽  
Gravitational Coupling ◽  
Dilaton Coupling

Using the renormalization group formalism, a sigma model of a special type — in which the metric and the dilaton depend explicitly on one of the string coordinates only — is investigated near two dimensions. It is seen that dilatonic gravity coupled to N scalar fields can be expressed in this form, using a string parametrization, and that it possesses the usual uv fixed point. However, in this stringy parametrization of the theory the fixed point for the scalar-dilaton coupling turns out to be trivial, while that for the gravitational coupling remains the same as in previous studies being, in particular, nontrivial.

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