On Dirichlet problem for second-order elliptic equations in the plane and uniform approximation problems for solutions of such equations

TL;DR

This paper investigates the regularity of Jordan domains with smooth boundaries for second-order elliptic equations, showing that such domains are generally not regular for these equations, and explores links to polynomial approximation problems.

Contribution

It demonstrates that smooth-boundary Jordan domains are not regular for non-strongly elliptic equations, highlighting limitations in boundary regularity and connecting to approximation theory.

Findings

Jordan domains with $C^{1,eta}$ boundary are not regular for non-strongly elliptic equations

Existence of Lipschitz boundary domains regular for bianalytic functions

Connections established between Dirichlet problem regularity and polynomial approximation

Abstract

We consider the Dirichlet problem for solutions to general second-order homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain with $C^{1,\alpha}$-smooth boundary, $0<\alpha<1$, is not regular with respect to the Dirichlet problem for any not strongly elliptic equation $\mathcal Lf=0$ of this kind, which means that for any such domain $G$ it always exists a continuous function on the boundary of $G$ that can not be continuously extended to the domain under consideration to a function satisfying the equation $\mathcal Lf=0$ therein. Since there exists a Jordan domain with Lipschitz boundary that is regular with respect to the Dirichlet problem for bianalytic functions, this result is near to be sharp. We also consider several connections between Dirichlet problem for elliptic equations under consideration and problems on uniform approximation by polynomial solutions of such equations.