Finite Element Approximation of Conformal Mappings to Unbounded Jordan Domains

In two-dimensional Euclidean space, a simply connected unbounded domain whose boundary is a Jordan curve is called an unbounded Jordan domain. In this article, we discuss a piecewise linear finite element approximation of conformal mappings from the unit disk of the plane to unbounded Jordan domains. Some convergence results and error analysis are presented. Numerical examples are also given.     

Quasiconformal harmonic mappings between \fancyscriptC1,m{{\fancyscript{C}}^{1,\mu}} Euclidean surfaces

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.     

On grid generation for numerical models of geophysical fluid dynamics

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for different spherical radii of geodesic disk. For a rectangular computational domain, the optimization problem is solved in the class of classical mappings and respective area of the spherical domain is evaluated.     

Some problems of conformal mappings of spherical domains

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.     

Some problems of conformal mappings of spherical domains

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.     

Convergence of Julia polynomials

We study the approximation of conformal mappings with the polynomials defined by Keldysh and Lavrentiev from an extremal problem considered by Julia. These polynomials converge uniformly on the closure of any Smirnov domain to the conformal mapping of this domain onto a disk. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angles at the boundary. Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.     

Grunsky inequalities and Fredholm spectrum in general domains

We discuss the Fredholm spectrum for general domains and study its applications to conformal and quasi-conformal mappings. In particular, we establish an improvement of the Grunsky inequalities which is valid for general domains. This improvement constitutes an extension of a recent result of Schiffer concerning the sharpening of Grunsky inequalities for the unit disk by a factor smaller than 1, and which is the reciprocal of the least Fredholm eigenvalue of the smooth simply-connected image domain.     

On the conformal mappings in the unit disk fixing arbitrarily many boundary points

In this paper, we give a method of constructing conformal mappings defined in the unit disk which can fix arbitrarily many points on the unit circle.     

Gears,pregears and related domains

We study conformal mappings from the unit disc to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or “gearlike”) domains fit into a more general category of domains we call “pregears” (images of gears under Möbius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.     

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

We study the Besov regularity of conformal mappings for domains with rough boundary based on the well-posedness for the Dirichlet problem with Besov data. Also, sharp invertibility results for the classical layer potential operators on Sobolev-Besov spaces on the boundary of curvilinear polygons are obtained.     

Two related free boundary problems

Two related free boundary problems are solved: the first isthe viscous film coating of wedges of arbitrary angle; the secondis the rectangular dam problem with evaporation (or fluid removal)from the free surface. Both problems are of practical interestand explicit solutions are given. The two examples treated aregeneralizations of problems solved using Polubarinova-Kochina's(1962) analytic differential equation method and conformal mappingsinvolving elliptic modular functions to an intermediate plane.Here conformal mappings involving Legendre functions are usedto generalize these results.     

Justification of the Method of Approximation of Solutions to Contact Problems of Filtration Theory

We suggest a new convergent algorithm for numerical solution of the nonlinear problem of finding the parameters of conformal mappings describing fluid filtration flows with free (contact) boundaries in porous media.     

General algorithms for split common fixed point problem of demicontractive mappings

In the first part of this paper, we present a new general algorithm for solving the split common fixed point problem for an infinite family of demicontractive mappings. We establish strong convergence of the algorithm in an infinite dimensional Hilbert space. As applications, we consider algorithms for split variational inequality problem and split common null point problem. In the second part of this paper, we present a new algorithm and strong convergence theorem for approximation of solutions of split equality fixed point problems for an infinite family of demicontractive mappings. Our results improve and generalize some recent results in the literature.     

Eigenfrequencies of fractal drums

A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in [L. Banjai, L.N. Trefethen, A multipole method for Schwarz–Christoffel mapping of polygons with thousands of sides, SIAM J. Sci. Comput. 25(3) (2003) 1042–1065]. The linear system arising from the spectral discretization is large and dense. To circumvent this problem we devise a fast method for the inversion of such a system. Consequently, the eigenvalue problem is solved iteratively. We obtain eight digits for the first eigenvalue of the Koch snowflake and at least five digits for eigenvalues up to the 20th. Numerical results for two more fractals are shown.     

A constructive method for starlike harmonic mappings

Summary The idea initiated by Opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings. This is of some interest, since such mappings are not characterized by analytic means.This work was supported in partsby a Promotion of Research Grant from the TECHNION, Haifaby an Undergraduate Student Research A ward from the NSERCby grants from the NSERC and the FCAR     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

Hadamard Products of Convex Harmonic Mappings

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \overline {g} $ where h and g are analytic functions. If $ f \in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \in K_H^0 $ . For $ f \in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\tilde {*}\varphi = h*\varphi + \overline {g*\varphi } $ is found. With } a strip mapping, $ f\tilde {*}\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \in K_H^0 $ which take the unit disk to regular n -gons.     

Constructing three-dimensional mappings onto the unit sphere with the hypercomplex Szegö kernel

In classical complex analysis the Szegö kernel method provides an explicit way to construct conformal maps from a given simply-connected domain G⊂C onto the unit disc. In this paper we revisit this method in the three-dimensional case. We investigate whether it is possible to construct three-dimensional mappings from some elementary domains into the three-dimensional unit ball by using the hypercomplex Szegö kernel. In the cases of rectangular domains, L-shaped domains, cylinders and the symmetric double-cone the proposed method leads surprisingly to qualitatively very good results. In the case of the cylinder we get even better results than those obtained by the hypercomplex Bergman method that was very recently proposed by several authors.We round off with also giving an explicit example of a domain, namely the T-piece, where the method does not lead to the desired result. This shows that one has to adapt the methods in accordance with different classes of domains.     

Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Gr?tzsch and Johannes C.C.?Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bj?rling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.