arXiv subjects

A. Rasila

Publications and source records attributed to A. Rasila.

3 recordsLinked to original sources

arXiv2025

Efficient Numerical Conformal Mappings on Multiply Connected Riemann Surfaces

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is generalized and refined to achieve the same level of accuracy on multiply connected planar domains and Riemann surfaces. The main challenge is the accurate and efficient construction of boundary values for the conjugate problem on multiply connected domains. The method relies on high-order finite element methods which allow for highly accurate computations of mappings on surfaces, including domains of complex boundary geometry containing strong singularities and cusps. We also derive the reciprocal error estimate for the multiply connected case. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments with error estimates.

math.NA↗

arXiv2013

On quasisymmetry of quasiconformal mappings and its applications

Suppose that $f: D\to D'$ is a quasiconformal mapping, where $D$ and $D'$ are domains in ${\mathbb R}^n$, and that $D$ is a broad domain. Then for every arcwise connected subset $A$ in $D$, the weak quasisymmetry of the restriction $f|_A: A\to f(A)$ implies its quasisymmetry, and as a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative under the additional condition that $A$ is arcwise connected. As an application, we establish nine equivalent conditions for a bounded domain, which is quasiconformally equivalent to a bounded and simply connected uniform domain, to be John. This result is a generalization of the main result of Heinonen from [Quasiconformal mappings onto John domains, \textit{Rev. Math. Iber.,} {\bf 5} (1989), 97--123].

math.CV↗

arXiv2012

Harmonic Close-to-convex Functions and Minimal Surfaces

In this paper, we study the family ${\mathcal C}_{H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{D}$ with $f_{\bar{z}}(0)=0$. We derive a sufficient condition for $f$ to belong to the class $\CC_{H}^0$. We take the analytic part of $f$ to be $zF(a,b;c;z)$ or $zF(a,b;c;z^2)$ and for a suitable choice of co-analytic part of $f$, the second complex dilatation $w(z)=\bar{f_{\bar{z}}}/f_z$ turns out to be a square of an analytic function. Hence $f$ is lifted to a minimal surface expressed by an isothermal parameter. Explicit representation for classes of minimal surfaces are given. Graphs generated by using Mathematica are used for illustration.

math.CV↗