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Categories: Harmonic analysis | Potential theory

Harmonic function

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 Laplace's equation, i.e.

$\frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0$

everywhere on U. This is also often written as

$\nabla^2 f = 0$ or

Δ f = 0.

There also exists a seemingly weaker definition that is equivalent. Indeed a function is harmonic if and only if it is weakly harmonic.

A function that satisfies $\Delta f \ge 0$ is said to be subharmonic.

Examples

Examples of harmonic functions are the constant, linear and affine functions on all of Rⁿ, the function

f(x₁, x₂) = ln(x₁² + x₂²)

on R² \ {0}, the function

f(x₁, x₂) = exp(x₁)sin(x₂),

and the function

f(x₁,...,x_n) = (x₁² + ... + x_n²)⁻ⁿ

on Rⁿ \ {0} if n ≥ 3.

Remarks

The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over R: sums, differences and scalar multiples of harmonic functions are again harmonic.

If f is a harmonic function on U, then all partial derivatives of f are also harmonic functions on U.

In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are analytic, i.e. they can be locally expressed as power series. This is a general fact about elliptic operators, of which the Laplacian is a major example.

The real and imaginary part of any holomorphic function yield harmonic functions on R². Conversely there is an operator taking a harmonic function u on a region in R² to its harmonic conjugate v, for which u+iv is a holomorphic function; here v is well-defined up to a real constant. This is well known in applications as (essentially) the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators . Geometrically u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which u and v are constant cross at right angles.

The harmonic functions satisfy the following maximum principle: if K is any compact subset of U, then f, restricted to K, attains its maximum and minimum on the boundary of K; there are no local maxima or minima, except if f is constant. If f is a harmonic function defined on all of Rⁿ which is bounded above or bounded below, then f is constant (compare Liouville's theorem). If B(x,r) is a ball with center x and radius r which is completely contained in U, then the value f(x) of the harmonic function f at the center of the ball is given by the average value of f on the surface of the ball; this average value is also equal to the average value of f in the interior of the ball.

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