Levi equation and local maximum property

TL;DR

This paper investigates the geometric properties of solutions to Dirichlet problems for the Levi operator on pseudoconvex domains in complex space, focusing on level sets and their hulls using the local maximum property.

Contribution

It introduces a novel approach to analyze the level sets of Levi operator solutions via hulls and extends techniques to study the Levi operator for graphs in complex two-space.

Findings

Characterization of level sets using hulls of intersections with boundary

Application of local maximum property to non-smooth solutions

Insights into Levi operator behavior for graphs in $\

Abstract

The aim of the paper is to study the level sets of the solutions of Dirichlet problems for the Levi operator on strongly pseudoconvex domains $\Omega$ in $\mathbb C^2$. Such solutions are generically non smooth, and the geometric properties of their level sets are characterized by means of hulls of their intersections with $b\Omega$, using as main tool the local maximum property introduced by Slodkowski (PJM, 1988). The same techniques are then employed to study the behavior of the complete Levi operator for graphs in $\mathbb C^2$.