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 x₀ inside the domain of f, there exist other points arbitrarily close to x₀ at which f takes larger values.

Mathematically, this can be formulated as follows. Let f be defined on some open subset D of the Euclidean space Rⁿ. If x₀ is a point in D such that

for all x in a neighborhood of x₀, 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.

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 x₀ 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 x₀ being a degenerate point, but this is the essential argument.

The maximum principle holds in more general circumstances. In fact, it is broadly speaking a property of elliptic operators.

See also

• Maximum modulus principle