Viscosity solutions to complex first eigenvalue equations

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.