The minimum sets and free boundaries of strictly plurisubharmonic functions

TL;DR

This paper investigates the structure and regularity of minimum sets of strictly plurisubharmonic functions with positive Monge-Ampère densities, revealing bounds on their Hausdorff dimension and analyzing free boundary problems with irregular data.

Contribution

It establishes bounds on the Hausdorff dimension of minimum sets, explores their relation to function regularity, and provides sharp examples and analogues in real analysis.

Findings

Minimum sets cannot contain large-dimensional analytic subvarieties.

Sharp examples illustrate the Hausdorff dimension limits of minimum sets.

Analysis of free boundary problems with irregular data in the planar case.

Abstract

We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Amp\`ere densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.