论文标题
复杂$ K $ -Hessian功能的Dirichlet原理
The Dirichlet principle for the complex $k$-Hessian functional
论文作者
论文摘要
我们研究有限域上的复杂$ k $ -Hessian方程的变异结构$ x \ subset \ mathbb c^n $带边界$ m = \ partial x $。我们证明dirichlet问题$σ_k（\ partial \ bar {\ partial} u）= 0 $ in $ x $，而$ m $上的$ u = f $ in $ m $是变量，我们对关联的功能$ \ Mathcal {e} _K（u）$进行了明确的构造。此外，我们证明$ \ Mathcal {e} _K（u）$满足Dirichlet原理。在$ k = 2 $的特殊情况下，我们构建的功能性$ \ Mathcal {e} _2（u）$涉及边界的Hermitian平均曲率，该概念首先由X. Wang引入和研究。 J. Case的早期工作和本文的第一作者介绍了（真实）$ K $ -Hessian功能的边界运营商，该功能满足了Dirichlet原理。本文表明，在复杂的环境中有一张平行的图片。
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 $σ_k (\partial \bar{\partial} u) =0$ 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.
