maximum principle
 Maximum principle

Let $f:U\to \mathbb{R}$ (where $U\subseteq {\mathbb{R}}^{d}$) be a harmonic function^{}. Then $f$ attains its extremal values on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$. If $f$ attains an extremal value anywhere in the interior of $K$, then it is constant.
 Maximal modulus principle

Let $f:U\to \u2102$ (where $U\subseteq \u2102$) be a holomorphic function^{}. Then $f$ attains its maximal value on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$. If $f$ attains its maximal value anywhere on the interior of $K$, then it is constant.
Title  maximum principle 
Canonical name  MaximumPrinciple 
Date of creation  20130322 12:46:06 
Last modified on  20130322 12:46:06 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  5 
Author  mathcam (2727) 
Entry type  Theorem 
Classification  msc 30C80 
Classification  msc 31A05 
Classification  msc 31B05 
Classification  msc 30F15 
Synonym  maximal modulus principle 
Synonym  maximum principle for harmonic functions 
Related topic  HadamardThreeCircleTheorem 
Related topic  PhragmenLindelofTheorem 