Continuity of Plurisubharmonic Envelopes in $\mathbb{C}^2$

TL;DR

This paper proves that certain smooth bounded pseudoconvex domains in ^2 are c-regular, meaning their plurisubharmonic envelopes of boundary-continuous functions are continuous, with applications to Reinhardt domains.

Contribution

It establishes the c-regularity of smooth bounded pseudoconvex domains in ^2 based on boundary regularity conditions, extending understanding of plurisubharmonic envelopes.

Findings

Strongly regular points form a closed set in the boundary.

Such domains are c-regular, ensuring continuity of plurisubharmonic envelopes.

Reinhardt domains in ^2 are c-regular.

Abstract

We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in $\mathbb{C}^{2}$ are $c$-regular.