Maximum principle

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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 x_0\, inside the domain of f, there exist other points arbitrarily close to x_0\, 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 x_0\, is a point in D such that

f(x_0)\ge f(x)

for all x in a neighborhood of x_0\,, 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 x_0\, 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 x_0\, 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.
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