On $L$-special domains with algebraic boundaries

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. We obtain new properties and examples of $L$-special domains with algebraic boundaries.