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 R^"n") 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:$abla^2\; f\; =\; 0$ or $Delta\; f\; =\; 0.$

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

Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace-de Rham operator$Delta$. In this context, a function is called "harmonic" if $Delta\; f\; =\; 0.$

A $C^2$ function that satisfies $Delta\; f\; ge\; 0$ is said to be "subharmonic".

Examples

Examples of harmonic functions of two variables are:
* the real and imaginary part of any holomorphic function
* the function ::"f"("x"₁, "x"₂) = ln("x"₁² + "x"₂²): defined on R² {0} (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass)
* the function "f"("x"₁, "x"₂) = exp("x"₁)sin("x"₂).

Examples of harmonic functions of "n" variables are:

*the constant, linear and affine functions on all of R^"n" (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
*the function "f"("x"₁,...,"x"_"n") = ("x"₁² + ... + "x"_"n"²)^{1 −"n"/2} on R^"n" {0} for "n" ≥ 2.

Examples of harmonic functions of three variables are given in the table below with $r^2=x^2+y^2+z^2$. Harmonic functions are determined by their singularities. The singular points of the harmonic functions below are expressed as "charges" and "charge densities" using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function below will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function.

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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". The Laplace operator Δ and the partial derivative operator will commute on this class of functions.

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 uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because any continuous function satisfying the mean value property is harmonic. Consider the sequence on ($-infty$, 0)× R defined by $scriptstyle\; f\_n(x,y)\; =\; frac1n\; exp(nx)cos(ny)$. This sequence is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This example shows the importance on relying on the mean value property and continuity to argue the limit is harmonic.

Connections with complex function theory

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. In this regard, "u+iv" would be the complex potential, where "u" is the potential function and "v" is the stream function.

Properties of harmonic functions

Some important properties of harmonic functions can be deduced from Laplace's equation.

The regularity theorem for harmonic functions

Harmonic functions are infinitely differentiable. In fact, harmonic functions are real analytic.

The maximum principle

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". If "U" is connected, this means that "f" cannot have local maxima or minima, other than the exceptional case where "f" is constant. Similar properties can be shown for subharmonic functions.

The mean value property

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. In other words

$u(x)\; =\; frac\{1\}\{nomega\_n\; r^\{n-1int\_\{partial\; B(x,r)\}\; u\; ,\; dS\; =\; frac\{1\}\{omega\_n\; r^n\}int\_\{B\; (x,r)\}\; u\; ,\; dV$

where $omega\_n$ is the volume of the unit ball in "n" dimensions.

Liouville's theorem

If "f" is a harmonic function defined on all of R^"n" which is bounded above or bounded below, then "f" is constant (compare Liouville's theorem for functions of a complex variable).

Generalizations

One generalization of the study of harmonic functions is the study of harmonic forms on Riemannian manifolds, and it is related to the study of cohomology. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as Dirichlet principle). These kind of harmonic maps appear in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in R to a Riemannian manifold, is a harmonic map if and only if it is a geodesic.

ee also

*Dirichlet problem
*Heat equation
*Laplace's equation
*Poisson's equation
*Quadrature domains
*Subharmonic function

References

* L.C. Evans, 1998. "Partial Differential Equations". American Mathematical Society.
*D. Gilbarg, N. Trudinger "Elliptic Partial Differential Equations of Second Order". ISBN 3-540-41160-7.
* Q. Han, F. Lin, 2000, "Elliptic Partial Differential Equations", American Mathematical Society

External links

*
* [http://math.fullerton.edu/mathews/c2003/HarmonicFunctionMod.html Harmonic Functions Module by John H. Mathews]
* [http://www.axler.net/HFT.html Harmonic Function Theory by S.Axler, Paul Bourdon, and Wade Ramey]