PlanetMath: maximum principle

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maximum principle

(Theorem)

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

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Other names:

maximal modulus principle, maximum principle for harmonic functions

Attachments:: proof of weak maximum principle for real domains (Proof) by rspuzio
proof of maximal modulus principle (Proof) by cvalente

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Cross-references: holomorphic function, interior, boundary, compact, harmonic function
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This is version 2 of maximum principle, born on 2002-06-09, modified 2004-10-23.
Object id is 3078, canonical name is MaximumPrinciple.
Accessed 11096 times total.

Classification:

AMS MSC:	30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination)
	31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	30F15 (Functions of a complex variable :: Riemann surfaces :: Harmonic functions on Riemann surfaces)

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