On $L$-special domains with algebraic boundaries

TL;DR

This work addresses the problem of uniform approximation on Carathéodory compacts by $L$-analytic polynomials, for second-order elliptic operators with constant complex coefficients in the plane. The authors reduce to the non-strongly elliptic case via a linear transform to $L_\beta$ and study when a Carathéodory domain is $L$-special by matching boundary data through $F_1 \in AC(D)$ and $F_2 \in AC(T_\beta D)$. They prove a degree constraint that for an $L$-special domain with boundary of order $n>2$, any admissible pair must satisfy $\max(\deg F_1, \deg F_2) > n$, and they construct a new explicit example with boundary order $4$, showing the existence of $D$ and an admissible polynomial pair $(F_1,F_2)$ with $F_1(z)=C z^5 - z$ and $F_2(z_\beta)=C \gamma^5 z_\beta^5 - \gamma z_\beta$ by implementing the operator $S_\beta$. The construction yields concrete parameters $\beta \approx 0.039$ and $\alpha \approx 3.96$ with the boundary close to the quartic curve $x^4+34.913 x^2 y^2+643.992 y^4=1$, and the paper notes the uniqueness of this example and open questions for higher-order boundaries. Overall, the results expand the catalog of $L$-special domains beyond ellipses and illuminate connections to the Dirichlet problem and uniform $L$-harmonic approximation.

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 Carathéodory 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.

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• Definition 2
• Lemma 3
• Theorem 3
• proof : Proof of Lemma \ref{['l1']}.
• proof : Proof of Lemma \ref{['l2']}.