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

In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminusΩ$. 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.

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