The exterior Dirichlet Problem for homogeneous complex $k$-Hessian equation

TL;DR

This paper addresses the exterior Dirichlet problem for the homogeneous complex k-Hessian equation in complex space, establishing existence, uniqueness, and regularity of solutions through key gradient and second order estimates.

Contribution

It provides the first proof of existence and uniqueness of $C^{1,1}$ solutions for the exterior complex k-Hessian problem with new uniform gradient and second order estimates.

Findings

Existence of $C^{1,1}$ solutions in exterior domains.

Uniqueness of solutions under given boundary conditions.

Development of uniform gradient and second order estimates.

Abstract

In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminus\Omega$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.