# 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 $f:U\to\mathbb{C}$ (where $U\subseteq\mathbb{C}$) 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