An example concerning Sadullaev's boundary relative extremal functions

TL;DR

This paper constructs a specific smoothly bounded domain demonstrating that Sadullaev's boundary relative extremal functions can satisfy strict inequalities, highlighting nuanced differences among these functions.

Contribution

It provides a concrete example showing that Sadullaev's boundary relative extremal functions can differ significantly, clarifying their relationships.

Findings

Existence of a smoothly bounded domain with specific boundary properties

Demonstration that $ ext{omega}_1(z,K, ext{Omega})< ext{omega}_2(z,K, ext{Omega})$

Clarification of inequalities among boundary relative extremal functions

Abstract

We exhibit a smoothly bounded domain $\Omega$ with the property that for suitable $K\subset\partial \Omega$ and $z\in \Omega$ the "Sadullaev boundary relative extremal functions" satisfy the inequality $\omega_1(z,K,\Omega)<\omega_2(z,K,\Omega)\le \omega(z,K,\Omega)$.