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.
everywhere on U. This is usually written as
or
Contents |
Examples of harmonic functions of two variables are:
Examples of harmonic functions of three variables are given in the table below with :
Function | Singularity |
---|---|
Unit point charge at origin | |
x-directed dipole at origin | |
Line of unit charge density on entire z-axis | |
Line of unit charge density on negative z-axis | |
Line of x-directed dipoles on entire z axis | |
Line of x-directed dipoles on negative z axis |
Finally, examples of harmonic functions of n variables are:
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 (, 0)× R defined by . 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 of relying on the mean value property and continuity to argue that the limit is harmonic.
The real and imaginary part of any holomorphic function yield harmonic functions on R2 (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function on an open set is locally the real part of a holomorphic function. This is immediately seen observing that, writing the complex function is holomorphic in because it satisfies the Cauchy-Riemann equations. Therefore, g has locally a primitive , and is the real part of up to a constant, as is the real part of .
Although the above correspondence with holomorphic functions only holds for functions of two real variables, still harmonic functions in n variables enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean-value principle; a theorem of removal of singularities as well as a Liouville theorem one holds for them in analogy to the corresponding theorems in complex functions theory.
Some important properties of harmonic functions can be deduced from Laplace's equation.
Harmonic functions are infinitely differentiable. In fact, harmonic functions are real analytic.
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.
If is a ball with center x and radius r which is completely contained in the open set , then the value of a harmonic function at the center of the ball is given by the average value of on the surface of the ball; this average value is also equal to the average value of in the interior of the ball. In other words
where is the volume of the unit ball in n dimensions and is the n-1 dimensional surface measure .
Conversely, all locally integrable functions satisfying the (volume) mean-value property are infinitely differentiable and harmonic functions as well.
In terms of convolutions, if
denotes the characteristic function of the ball with radius r about the origin, normalized so that , the function is harmonic on if and only if
as soon as
Sketch of the proof. The proof of the mean-value property of the harmonic functions and its converse follows immediately observing that the non-homogeneous equation, for any
admits an easy explicit solution of class with compact support in .Thus, if is harmonic in
holds in the set of all points with .
Since is continuous in converges to as showing the mean value property for in .Conversely, if u is any function satisfying the mean-value property in , that is,
holds in for all then, iterating m times the convolution with one has:
so that is because the m-fold iterated convolution of is of class with support .Since and are arbitrary, is too. Moreover
for all so that in 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 ∫h = 1, then u(x) = h * u(x). In other words, we can take the weighted average of u about a point and recover u(x). 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. See Weyl's lemma.
Let u be a non-negative harmonic function in a bounded domain Ω. Then for every connected set
holds for some constant C that depends only on V and Ω.
The following principle of removal of singularities holds for harmonic functions. If f is a harmonic function defined on a dotted open subset of Rn, which is less singular at than the fundamental solution, that is
then f extends to a harmonic function on (compare Riemann's theorem for functions of a complex variable).
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 for functions of a complex variable).
Edward Nelson gave a particularly short proof [1] of this theorem, 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.
A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation
in a weak sense (or, equivalently, in the sense of distributions). A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is Weyl's lemma.
There are other weak formulations of Laplace's equation that are often useful. One of which is Dirichlet's principle, representing harmonic functions in the Sobolev space H1(Ω) as the minimizers of the Dirichlet energy integral
with respect to local variations, that is, all functions such that holds for all or equivalently, for all
Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace-Beltrami operator Δ. In this context, a function is called harmonic if
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.
A C2 function that satisfies is called subharmonic. This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail. More generally, a function is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball.
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.
If M and N are two Riemannian manifolds, then a harmonic map u : M → N is defined to be a stationary point of the Dirichlet energy
in which du : TM → TN 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 T∗M⊗u−1TN.
Important special cases of harmonic maps between manifolds include minimal surfaces, which are precisely the harmonic immersions of a surface into three-dimensional Euclidean space. More generally, minimal submanifolds are harmonic immersions of one manifold in another. Harmonic coordinates are a harmonic diffeomorphism from a manifold to an open subset of a Euclidean space of the same dimension.