发表机构
Baylor University; Temple University（贝勒大学; 天普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Agmon猜想对任意阶多调和算子成立，通过引入新型多层奇异积分算子，在粗糙区域中解决了高阶椭圆方程的Dirichlet问题。
AI 中文摘要
在1957年关于平面中高阶椭圆方程Dirichlet问题的论文中，Agmon猜想在维数大于二的空间中应存在高阶方程，使得基于多层积分算子的位势理论方法能够成功实施。我们证明，对于$\u0052^n$（$n\geq 2$）中任意阶$m\in\mathbb{N}$的多调和算子$\Delta^m$，即使在允许超出现有高阶椭圆边值问题理论范围的奇异性的几何粗糙区域中，这一猜想确实成立。我们方法的核心是一类新型的多层奇异积分算子，它们对于$\Delta^m$所起的作用，相当于经典调和双层位势在处理调和函数Dirichlet问题时所扮演的角色。
英文摘要
In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this is indeed the case for the polyharmonic operator $Δ^m$ of arbitrary order $m\in\mathbb{N}$ in $\mathbb{R}^n$, $n\geq 2$, even in geometrically rough domains permitting singularities beyond the scope of the existing theory of higher-order elliptic boundary value problems. Central to our approach is a new genre of multi-layer singular integral operators, which play for $Δ^m$ the role assumed by the classical harmonic double layer potential in the treatment of the Dirichlet problem for harmonic functions.
