Viscosity solutions to complex first eigenvalue equations

TL;DR

This paper investigates viscosity solutions to complex first eigenvalue equations in bounded domains, establishing existence, uniqueness, and extending the theory to related operators in several complex variables.

Contribution

It proves the existence and uniqueness of viscosity solutions for the complex first eigenvalue problem and extends viscosity theory to comparable operators.

Findings

Unique viscosity solutions exist for the Dirichlet problem in bounded B-regular domains.

The theory is extended to operators comparable to the first eigenvalue operator.

Results contribute to the understanding of complex eigenvalue equations in several complex variables.

Abstract

We study the viscosity solutions to the first eigenvalue equation. We consider $\Omega$ a bounded B-regular domain in $\mathbb{C}^n$ and we prove that the Dirichlet problem $\Lambda_{1}(D_{\mathbb{C}}^2 u)=f$ in $\Omega$ and $u=\varphi$ on $\partial\Omega$ admits a unique viscosity solution. We also deal with viscosity theory for operators which are comparable to the first eigenvalue operator.