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 of C-n with boundary M=partial derivative X. We prove that the Dirichlet problem sigma(k)(partial derivative partial derivative<overline>u)=0 in X, and u=f on M is variational and we give an explicit construction of the associated functional epsilon(k)(u). Moreover we prove epsilon(k)(u) satisfies the Dirichlet principle. In a special case when k=2, our constructed functional epsilon(2)(u) involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang. Earlier work of J. Case and and the first author of this article 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.