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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

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 In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

for all x in a neighborhood of x₀, then the function f is constant on D. This is a glossary of some terms used in the branch of mathematics known as topology. ...

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. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... Chemistry In chemistry, a critical point is the conditions ( temperature, pressure) at which the liquid state of the matter ceases to exist. ... Saddle point in the graph of z=x²-y² In mathematics, a saddle point is a point of a function of two variables which is a stationary point but not a local extremum. ...

The maximum principle holds in more general circumstances. In fact, it is broadly speaking a property of elliptic operators. In mathematics, an Elliptic operator is a major type of differential operator P defined on spaces of complex-valued functions, or some more general function-like objects, such that the coefficients of the highest-order derivatives satisfy a positivity condition. ...

See also

• Maximum modulus principle