Harmonic function

The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.

Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics.

On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity.

All harmonic functions are analytic, that is, they can be locally expressed as power series.

This is a general fact about elliptic operators, of which the Laplacian is a major example.

This is true because every continuous function satisfying the mean value property is harmonic.

This example shows the importance of relying on the mean value property and continuity to argue that the limit is harmonic.

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

Harmonic functions satisfy the following maximum principle: if K is a nonempty 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.

If B(x, r) is a ball with center x and radius r which is completely contained in the open set

where ωn is the volume of the unit ball in n dimensions and σ is the (n − 1)-dimensional surface measure.

Conversely, all locally integrable functions satisfying the (volume) mean-value property are both infinitely differentiable and harmonic.

denotes the characteristic function of the ball with radius r about the origin, normalized so that

The proof of the mean-value property of the harmonic functions and its converse follows immediately observing that the non-homogeneous equation, for any 0 < s < r

admits an easy explicit solution wr,s of class C1,1 with compact support in B(0, r).

for all 0 < s < r so that Δu = 0 in Ω by the fundamental theorem of the calculus of variations, proving the equivalence between harmonicity and mean-value property.

This statement of the mean value property can be generalized as follows: If h is any spherically symmetric function supported in B(x, r) such that

In particular, by taking h to be a C∞ function, we can recover the value of u at any point even if we only know how u acts as a distribution.

Edward Nelson gave a particularly short proof of this theorem for the case of bounded functions,[2] using the mean value property mentioned above: Given two points, choose two balls with the given points as centers and of equal radius.

If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume.

Since f is bounded, the averages of it over the two balls are arbitrarily close, and so f assumes the same value at any two points.

In the last expression, we may multiply and divide by vol Br and use the averaging property again, to obtain

In words, it says that a harmonic function defines a martingale for the Brownian motion.

[3] A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation

Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace–Beltrami operator Δ.

Many of the properties of harmonic functions on domains in Euclidean space carry over to this more general setting, including the mean value theorem (over geodesic balls), the maximum principle, and the Harnack inequality.

With the exception of the mean value theorem, these are easy consequences of the corresponding results for general linear elliptic partial differential equations of the second order.

This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail.

This kind of harmonic map appears in the theory of minimal surfaces.

is the differential of u, and the norm is that induced by the metric on M and that on N on the tensor product bundle