Very weak solutions of quadratic Hessian equations

TL;DR

This paper extends methods to study very weak solutions of quadratic Hessian equations in complex domains, improving understanding of regularity thresholds and broadening the scope of solution concepts in complex analysis.

Contribution

It introduces new techniques for analyzing very weak solutions of the complex  equation and refines the regularity thresholds for counterexamples in the real case.

Findings

Extended methods to complex  equations

Sharpened regularity thresholds for counterexamples

Broadened understanding of weak solutions in complex analysis

Abstract

We extend the methods of Lewicka - Pakzad, Sz\'ekelyhidi - Cao and Li - Qiu to study the notion of very weak solutions to the complex $\sigma_2$ equation in domains in $\mathbb C^n,\ n\geq 2$. As a by-product we sharpen the regularity threshold of the counterexamples obtained by Li and Qiu in the real case.