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

TL;DR

This paper establishes a variational framework for the complex $k$-Hessian equation on bounded domains, constructing an explicit functional that satisfies the Dirichlet principle, extending previous real-variable results to the complex setting.

Contribution

It introduces a variational structure and explicit functional for the complex $k$-Hessian equation, including a special case involving Hermitian mean curvature.

Findings

The Dirichlet problem for the complex $k$-Hessian equation is variational.

An explicit functional $\\mathcal{E}_k(u)$ satisfying the Dirichlet principle is constructed.

For $k=2$, the functional involves the Hermitian mean curvature of the boundary.

Abstract

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. 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.