Viscosity solutions to complex first eigenvalue equations

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

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