Harmonic function

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File:Laplace's equation on an annulus.svg

A harmonic function defined on an annulus.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function <math>f\colon U \to \mathbb R,</math> where Template:Mvar is an open subset of Template:Tmath that satisfies Laplace's equation, that is, <math display="block"> \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math> everywhere on Template:Mvar. This is usually written as <math display="block"> \nabla^2 f = 0 </math> or <math display="block">\Delta f = 0</math>

Etymology of the term "harmonic"Edit

The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. 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. Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmonic" was used to refer to all functions satisfying Laplace's equation.<ref>Template:Cite book</ref>

ExamplesEdit

Examples of harmonic functions of two variables are:

The real or imaginary part of any holomorphic function.
The function <math>\,\! f(x, y) = e^{x} \sin y;</math> this is a special case of the example above, as <math>f(x, y) = \operatorname{Im}\left(e^{x+iy}\right) ,</math> and <math>e^{x+iy}</math> is a holomorphic function. The second derivative with respect to x is <math>\,\! e^{x} \sin y,</math> while the second derivative with respect to y is <math>\,\! -e^{x} \sin y.</math>
The function <math>\,\! f(x, y) = \ln \left(x^2 + y^2\right)</math> defined on <math>\mathbb{R}^2 \smallsetminus \lbrace 0 \rbrace .</math> This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass.

Examples of harmonic functions of three variables are given in the table below with <math>r^2=x^2+y^2+z^2:</math>

Function	Singularity
<math>\frac{1}{r}</math>	Unit point charge at origin
<math>\frac{x}{r^3}</math>	x-directed dipole at origin
<math>-\ln\left(r^2 - z^2\right)\,</math>	Line of unit charge density on entire z-axis
<math>-\ln(r + z)\,</math>	Line of unit charge density on negative z-axis
<math>\frac{x}{r^2 - z^2}\,</math>	Line of x-directed dipoles on entire z axis
<math>\frac{x}{r(r + z)}\,</math>	Line of x-directed dipoles on negative z axis

Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). 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. In this case, uniqueness follows by Liouville's theorem.

The singular points of the harmonic functions above 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 above 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.

Finally, examples of harmonic functions of Template:Mvar variables are:

The constant, linear and affine functions on all of Template:Tmath (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
The function <math> f(x_1, \dots, x_n) = \left({x_1}^2 + \cdots + {x_n}^2\right)^{1-n/2}</math> on <math>\mathbb{R}^n \smallsetminus \lbrace 0 \rbrace</math> for Template:Math.

PropertiesEdit

The set of harmonic functions on a given open set Template:Mvar can be seen as the kernel of the Laplace operator Template:Math and is therefore a vector space over Template:Tmath linear combinations of harmonic functions are again harmonic.

If Template:Mvar is a harmonic function on Template:Mvar, then all partial derivatives of Template:Mvar are also harmonic functions on Template:Mvar. The Laplace operator Template:Math 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, 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.

The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider the sequence on Template:Tmath defined by <math display="inline">f_n(x,y) = \frac 1 n \exp(nx)\cos(ny);</math> 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.

Connections with complex function theoryEdit

The real and imaginary part of any holomorphic function yield harmonic functions on Template:Tmath (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function Template:Mvar on an open subset Template:Math of Template:Tmath is locally the real part of a holomorphic function. This is immediately seen observing that, writing <math>z = x + iy,</math> the complex function <math>g(z) := u_x - i u_y</math> is holomorphic in Template:Math because it satisfies the Cauchy–Riemann equations. Therefore, Template:Mvar locally has a primitive Template:Mvar, and Template:Mvar is the real part of Template:Mvar up to a constant, as Template:Mvar is the real part of <math>f' = g.</math>

Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in Template:Mvar variables still 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 holds for them in analogy to the corresponding theorems in complex functions theory.

Properties of harmonic functionsEdit

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

Regularity theorem for harmonic functionsEdit

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

Maximum principleEdit

Harmonic functions satisfy the following maximum principle: if Template:Mvar is a nonempty compact subset of Template:Mvar, then Template:Mvar restricted to Template:Mvar attains its maximum and minimum on the boundary of Template:Mvar. If Template:Mvar is connected, this means that Template:Mvar cannot have local maxima or minima, other than the exceptional case where Template:Mvar is constant. Similar properties can be shown for subharmonic functions.

The mean value propertyEdit

If Template:Math is a ball with center Template:Mvar and radius Template:Mvar which is completely contained in the open set <math>\Omega \subset \R^n,</math> then the value Template:Math of a harmonic function <math>u: \Omega \to \R</math> at the center of the ball is given by the average value of Template:Mvar on the surface of the ball; this average value is also equal to the average value of Template:Mvar in the interior of the ball. In other words, <math display="block">u(x) = \frac{1}{n\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{1}{\omega_n r^n}\int_{B(x,r)} u\, dV</math> where Template:Mvar is the volume of the unit ball in Template:Mvar dimensions and Template:Mvar is the Template:Math-dimensional surface measure.

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

In terms of convolutions, if <math display="block">\chi_r := \frac{1}{|B(0, r)|}\chi_{B(0, r)} = \frac{n}{\omega_n r^n}\chi_{B(0, r)}</math> denotes the characteristic function of the ball with radius Template:Mvar about the origin, normalized so that <math display="inline">\int_{\R^n}\chi_r\, dx = 1,</math> the function Template:Mvar is harmonic on Template:Math if and only if <math display="block">u(x) = u*\chi_r(x)\;</math> for all x and r such that <math>B(x,r) \subset \Omega.</math>

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 Template:Math <math display="block">\Delta w = \chi_r - \chi_s\;</math> admits an easy explicit solution Template:Mvar of class Template:Math with compact support in Template:Math. Thus, if Template:Mvar is harmonic in Template:Math <math display="block">0=\Delta u * w_{r,s} = u*\Delta w_{r,s}= u*\chi_r - u*\chi_s\;</math> holds in the set Template:Math of all points Template:Mvar in Template:Math with <math>\operatorname{dist}(x,\partial\Omega) > r.</math>

Since Template:Mvar is continuous in Template:Math, <math>u * \chi_s</math> converges to Template:Mvar as Template:Math showing the mean value property for Template:Mvar in Template:Math. Conversely, if Template:Mvar is any <math>L^1_{\mathrm{loc}}\;</math> function satisfying the mean-value property in Template:Math, that is, <math display="block">u*\chi_r = u*\chi_s\;</math> holds in Template:Math for all Template:Math then, iterating Template:Mvar times the convolution with Template:Math one has: <math display="block">u = u*\chi_r = u*\chi_r*\cdots*\chi_r\,,\qquad x\in\Omega_{mr},</math> so that Template:Mvar is <math>C^{m-1}(\Omega_{mr})\;</math> because the Template:Mvar-fold iterated convolution of Template:Math is of class <math>C^{m-1}\;</math> with support Template:Math. Since Template:Mvar and Template:Mvar are arbitrary, Template:Mvar is <math>C^{\infty}(\Omega)\;</math> too. Moreover, <math display="block">\Delta u * w_{r,s} = u*\Delta w_{r,s} = u*\chi_r - u*\chi_s = 0\;</math> for all Template:Math so that Template:Math in Template:Math 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 Template:Mvar is any spherically symmetric function supported in Template:Math such that <math display="inline">\int h = 1,</math> then <math>u(x) = h * u(x).</math> In other words, we can take the weighted average of Template:Mvar about a point and recover Template:Math. In particular, by taking Template:Mvar to be a Template:Math function, we can recover the value of Template:Mvar at any point even if we only know how Template:Mvar acts as a distribution. See Weyl's lemma.

Harnack's inequalityEdit

Let <math display="block">V \subset \overline{V} \subset \Omega</math> be a connected set in a bounded domain Template:Math. Then for every non-negative harmonic function Template:Mvar, Harnack's inequality <math display="block">\sup_V u \le C \inf_V u</math> holds for some constant Template:Mvar that depends only on Template:Mvar and Template:Math.

Removal of singularitiesEdit

The following principle of removal of singularities holds for harmonic functions. If Template:Mvar is a harmonic function defined on a dotted open subset <math>\Omega \smallsetminus \{x_0\}</math> of Template:Tmath, which is less singular at Template:Math than the fundamental solution (for Template:Math), that is <math display="block">f(x)=o\left( \vert x-x_0 \vert^{2-n}\right),\qquad\text{as }x\to x_0,</math> then Template:Mvar extends to a harmonic function on Template:Math (compare Riemann's theorem for functions of a complex variable).

Liouville's theoremEdit

Theorem: If Template:Mvar is a harmonic function defined on all of Template:Tmath which is bounded above or bounded below, then Template:Mvar is constant.

(Compare Liouville's theorem for functions of a complex variable).

Edward Nelson gave a particularly short proof of this theorem for the case of bounded functions,<ref>Template:Cite journal</ref> 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 Template:Mvar is bounded, the averages of it over the two balls are arbitrarily close, and so Template:Mvar assumes the same value at any two points.

The proof can be adapted to the case where the harmonic function Template:Mvar is merely bounded above or below. By adding a constant and possibly multiplying by –1, we may assume that Template:Mvar is non-negative. Then for any two points Template:Mvar and Template:Mvar, and any positive number Template:Mvar, we let <math>r=R+d(x,y).</math> We then consider the balls Template:Math and Template:Math where by the triangle inequality, the first ball is contained in the second.

By the averaging property and the monotonicity of the integral, we have <math display="block">f(x)=\frac{1}{\operatorname{vol}(B_R)}\int_{B_R(x)}f(z)\, dz\leq \frac{1}{\operatorname{vol}(B_R)} \int_{B_r(y)}f(z)\, dz.</math> (Note that since Template:Math is independent of Template:Mvar, we denote it merely as Template:Math.) In the last expression, we may multiply and divide by Template:Math and use the averaging property again, to obtain <math display="block">f(x)\leq \frac{\operatorname{vol}(B_r)}{\operatorname{vol}(B_R)}f(y).</math> But as <math>R\rightarrow\infty ,</math> the quantity <math display="block">\frac{\operatorname{vol}(B_r)}{\operatorname{vol}(B_R)} = \frac{\left(R+d(x,y)\right)^n}{R^n}</math> tends to 1. Thus, <math>f(x)\leq f(y).</math> The same argument with the roles of Template:Mvar and Template:Mvar reversed shows that <math>f(y)\leq f(x)</math>, so that <math>f(x) = f(y).</math>

Another proof uses the fact that given a Brownian motion Template:Mvar in Template:Tmath such that <math>B_0 = x_0,</math> we have <math>E[f(B_t)] = f(x_0)</math> for all Template:Math. In words, it says that a harmonic function defines a martingale for the Brownian motion. Then a probabilistic coupling argument finishes the proof.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

GeneralizationsEdit

Weakly harmonic functionEdit

A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation <math display="block">\Delta f = 0\,</math> 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 Template:Math as the minimizers of the Dirichlet energy integral <math display="block">J(u):=\int_\Omega |\nabla u|^2\, dx</math> with respect to local variations, that is, all functions <math>u\in H^1(\Omega)</math> such that <math>J(u) \leq J(u+v)</math> holds for all <math>v\in C^\infty_c(\Omega),</math> or equivalently, for all <math>v\in H^1_0(\Omega).</math>

Harmonic functions on manifoldsEdit

Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace–Beltrami operator Template:Math. In this context, a function is called harmonic if <math display="block">\ \Delta f = 0.</math> 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.

Subharmonic functionsEdit

A Template:Math function that satisfies Template:Math 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.

Harmonic formsEdit

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). This kind of harmonic map appears in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in Template:Tmath to a Riemannian manifold, is a harmonic map if and only if it is a geodesic.

Harmonic maps between manifoldsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If Template:Mvar and Template:Mvar are two Riemannian manifolds, then a harmonic map <math>u: M \to N</math> is defined to be a critical point of the Dirichlet energy <math display="block">D[u] = \frac{1}{2} \int_M \left\|du\right\|^2 \, d\operatorname{Vol}</math> in which <math>du: TM \to TN </math> is the differential of Template:Mvar, and the norm is that induced by the metric on Template:Mvar and that on Template:Mvar on the tensor product bundle <math>T^\ast M \otimes u^{-1} TN.</math>

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.