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maximum principle (Theorem)
Maximum principle
Let $f:U\to \Reals$ (where $U\subseteq \Reals^d$ ) 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 $f:U\to\Complex$ (where $U\subseteq \Complex$ ) 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.




"maximum principle" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: Hadamard three-circle theorem, Phragmén-Lindelöf theorem

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
There are 5 references to this entry.

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 13124 times total.

Classification:
AMS MSC30C80 (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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