On nonequivalence of regular boundary points for second-order elliptic operators

TL;DR

This paper constructs examples of second-order elliptic operators with continuous or discontinuous coefficients that demonstrate irregular boundary points, highlighting differences from the Laplacian's boundary regularity in various dimensions.

Contribution

It provides explicit examples showing the nonequivalence of boundary point regularity between certain elliptic operators and the Laplacian, including cases with continuous and discontinuous coefficients.

Findings

Existence of irregular boundary points for some elliptic operators with continuous coefficients.

Operators with discontinuous coefficients can have boundary points regular for Laplacian but not for the operator.

Examples constructed in each dimension starting from 3.

Abstract

In this paper we present examples of nondivergence form second order elliptic operators with continuous coefficients such that $L$ has an irregular boundary point that is regular for the Laplacian. Also for any eigenvalue spread <1 of the matrix of the coefficients we provide an example of operator with discontinuous coefficients that has regular boundary points nonequivalent to Laplacian's (we give examples for each direction of nonequivalence). All examples are constructed for each dimension starting with 3.