论文标题
复杂的Hessian方程的连续订阅问题
The Continuous Subsolution Problem for Complex Hessian Equations
论文作者
论文摘要
令$ω\ subset \ mathbb c^n $为严格的$ m $ -M $ -PSEUDOCONVEX域（$ 1 \ leq m \ leq n $）和$μ$ $ $ $ a $ ch. $ω$。 我们研究了复杂的Hessian方程$（dd^c u）^m \wedgeβ^{n -m} =μ$上的dirichlet问题。 首先，我们就$ m $ - 赫西亚的容量提供了$ m $ $ $的“扩散模量”的足够条件，该容量可以保证使用连续的边界数据与相关的dirichlet问题的连续解决方案。 作为一个应用程序，我们证明，如果方程具有连续的$ m $ -subharmonic subsustoluts，其连续性模量满足DINI类型条件，则该方程具有带有任意连续边界数据的连续解决方案。此外，当该度量在$ω$上具有有限质量时，我们对解决方案连续性的模量进行了精确的定量估计。 我们证明的主要步骤之一是建立一个新的容量估计值，提供对$ m $ -Hessian的扩散模量的准确估计，以$ m $ - $ m $ - $ m $ - $ m $ - $ m $ - $ m $ - $ m $ -Hessian的能力在$ CONTUNULUS上的连续符合$ C的连续性。 另一个重要的成分是$ m $ -HESSIAN的新弱稳定性估计值，用于$ω$中的连续$ m $ $ m $ -subharmonic功能。
Let $Ω\subset \mathbb C^n$ be a bounded strictly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure on $Ω$. We study the Dirichlet problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. First we give a sufficient condition on the "modulus of diffusion" of the measure $μ$ with respect to the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass on $Ω$, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in our proof is to establish a new capacity estimate providing a precise estimate of the modulus of diffusion of the $m$-Hessian measure of a continuous $m$-subharmonic function $φ$ in $Ω$ with zero boundary with respect to the $m$-Hessian capacity in terms of the modulus of continuity of $φ$. Another important ingredient is a new weak stability estimate for the $m$-Hessian measure of a continuous $m$-subharmonic function in $Ω$.
