V2EX superharmonic

Superharmonic

释义 Definition

superharmonic(超调和的):在数学(尤其是势论/偏微分方程)中,指一种函数性质:它通常满足“平均值不等式”(在某区域内,函数在一点的值不小于其在任意小球面/小球上的平均值),直观上表示“不会在内部产生比边界更低的‘凹陷’”。最常见用法是 superharmonic function(超调和函数)
(在某些工程/声学语境中也可能指“高于基频谐波的”,但数学义更常见。)

发音 Pronunciation

/suprhrmnk/

例句 Examples

A superharmonic function cannot have a strict local minimum inside the domain.
超调和函数不可能在定义域内部出现严格的局部最小值。

In potential theory, superharmonic functions help describe equilibrium states and are linked to Laplace’s equation through mean-value inequalities.
在势论中,超调和函数有助于描述平衡状态,并通过平均值不等式与拉普拉斯方程建立联系。

词源 Etymology

super-(“在上、超过”)+ harmonic(“调和的;与调和函数相关的”)构成。这里的 harmonic 源于数学中的 harmonic function(调和函数),常与拉普拉斯算子相关;superharmonic 则表示在该框架下“更强/更上界”的一类函数性质(通常表现为不等式形式)。

相关词 Related Words

文献与作品 Literary Works

  • Ransford, Potential Theory in the Complex Plane(讨论超调和/次调和函数在复平面势论中的作用)
  • Helms, Introduction to Potential Theory(系统介绍势论中超调和函数的基本性质)
  • Evans, Partial Differential Equations(在PDE与势论相关章节中涉及超调和函数与最大值原理思想)
  • Armitage & Gardiner, Classical Potential Theory(经典势论语境下的超调和函数与相关工具)
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