maximum principle

Maximum principle

Let $f:U\to ℝ$ (where $U\subseteq {ℝ}^d$) 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 $f:U\to ℂ$ (where $U\subseteq ℂ$) 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	2013-03-22 12:46:06
Last modified on	2013-03-22 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