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 $u=\phi$ on $\partial \Omega$ admits a unique viscosity solution. We also deal with viscosity theory for operators which are comparable to the first eigenvalue operator.

arXiv commentWhen this paper got published, Reese Harvey and Blaine Lawson informed the author that the main result of this paper follows from their work: Harvey, F. Reese; Lawson, H. Blaine Jr. The Inhomogeneous Dirichlet problem for natural operators on manifolds. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3017-3064.pp