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 $\href{https://doi.org/10.1142/S0219199716500632}{[37]}$. Earlier work of J. Case and and the first author of this article $\href{https://doi.org/10.1016/j.jfa.2018.08.024}{[9]}$ 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.

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Received 1 June 2021

Accepted 10 June 2021

Published 13 August 2024