Maximum principle
From Wikipedia, the free encyclopedia
- This article describes the maximum principle in mathematics. For the maximum principle in optimal control theory, see Pontryagin's minimum principle.
In mathematics, the maximum principle in harmonic analysis states that if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. In other words, either f is a constant function, or, for any point 
inside the domain of f, there exist other points arbitrarily close to at which f takes larger values.
This principle can be formulated mathematically as follows. Let f be defined on some connected open subset D of the Euclidean space Rn. If is a point in D such that
for all x in a neighborhood of , then the function f is constant on D.
By replacing "maximum" with "minimum" and "larger" with "smaller", one obtains the minimum principle for harmonic functions.
The maximum principle also holds for the more general subharmonic functions, while superharmonic functions satisfy the minimum principle.
[edit] Heuristics behind the proof
The key ingredient for the proof is the fact that, by the definition of a harmonic function, the Laplacian of f is zero. Then, if is a non-degenerate critical point of f(x), we must be seeing a saddle point, since otherwise there is no chance that the sum of the second derivatives of f is zero. This of course is not a complete proof, and we left out the case of being a degenerate point, but this is the essential idea.
The maximum principle holds in more general circumstances. In fact, it is broadly speaking a property of elliptic operators.
[edit] See also
[edit] References
- Carlos A. Berenstein and Roger Gay, Complex Variables : An Introduction (Graduate Texts in Mathematics). Springer; 1 edition (1997). ISBN 0-387-97349-4.
