On $L$-special domains with algebraic boundaries

TL;DR

This paper explores properties and examples of $L$-special domains with algebraic boundaries, linking complex analysis and elliptic PDEs, and advances understanding of their approximation characteristics.

Contribution

It introduces new properties and examples of $L$-special domains with algebraic boundaries, enriching the theory and applications of these domains.

Findings

New properties of $L$-special domains identified

Examples of algebraic boundary domains constructed

Connections to polynomial approximation in elliptic PDEs established

Abstract

The concept of $L$-special domain appeared in the early 2000s. This analytical characteristic of domains in the complex plane is related to the problem on uniform approximation of functions on Carath\'eodory compacts in $\mathbb{R}^2$ by polynomial solutions of homogeneous second-order elliptic partial differential equations $Lu=0$ with constant complex coefficients. In this paper, new properties and examples of $L$-special domains with algebraic boundaries are obtained.