The Dirichlet principle for the complex $k$-Hessian functional

We study the variational structure of the complex $k$-Hessian equation on bounded domain $X\subset \mathbb C^{n}$ with boundary $M=\partial X$. We prove that the Dirichlet problem $\sigma _{k} (\partial \bar{\partial} u) =0$ in $X$, and $u=f$ on $M$ is variational and we give an explicit construction of the associated functional $ \mathcal{E}_{k}(u)$. Moreover we prove $ \mathcal{E}_{k}(u)$ satisfies the Dirichlet principle. In a special case when $k=2$, our constructed functional $ \mathcal{E}_{2}(u)$ involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang [37]. Earlier work of J. Case and and the first author of this article [9] introduced a boundary operator for the (real) $k$-Hessian functional which satisfies the Dirichlet principle. The present paper shows that there is a parallel picture in the complex setting.