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}, that satisfies Laplace's equation, that is,

- Etymology of the term "harmonic"
- Examples
- Remarks
- Connections with complex function theory
- Properties of harmonic functions
- Regularity theorem for harmonic functions
- Maximum principle
- The mean value property
- Harnack's inequality
- Removal of singularities
- Liouville's theorem
- Generalizations
- Weakly harmonic function
- Harmonic functions on manifolds
- Subharmonic functions
- Harmonic forms
- Harmonic maps between manifolds
- See also
- Notes
- References
- External links

everywhere on *U*. This is usually written as

or

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.^{ [1] }

Examples of harmonic functions of two variables are:

- The real and imaginary parts of any holomorphic function
- The function ; this is a special case of the example above, as , and is a holomorphic function.
- The function defined on . 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 :

Function Singularity Unit point charge at origin *x*-directed dipole at originLine 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*axisLine 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 *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 on for
*n*> 2.

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**: linear combinations 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, 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 (−∞, 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 **R**^{2} (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function *u* on an open subset Ω of **R**^{2} is *locally* the real part of a holomorphic function. This is immediately seen observing that, writing *z* = *x* + *iy*, the complex function *g*(*z*) := *u _{x}* − i

Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in *n* 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.

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

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

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. Similar properties can be shown for subharmonic functions.

If *B*(*x*, *r*) is a ball with center *x* and radius *r* which is completely contained in the open set Ω ⊂ **R**^{n}, then the value *u*(*x*) of a harmonic function *u*: Ω → **R** at the center of the ball is given by the average value of *u* on the surface of the ball; this average value is also equal to the average value of *u* in the interior of the ball. In other words,

where *ω*_{n} is the area of the unit sphere 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.

In terms of convolutions, if

denotes the characteristic function of the ball with radius *r* about the origin, normalized so that , the function *u* is harmonic on Ω if and only if

as soon as *B*(*x*, *r*) ⊂ Ω.

**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 0 < *s* < *r*

admits an easy explicit solution *w _{r,s}* of class

holds in the set Ω_{r} of all points *x* in with .

Since *u* is continuous in Ω, *u**χ_{r} converges to *u* as *s* → 0 showing the mean value property for *u* in Ω. Conversely, if *u* is any function satisfying the mean-value property in Ω, that is,

holds in Ω_{r} for all 0 < *s* < *r* then, iterating *m* times the convolution with χ_{r} one has:

so that *u* is because the m-fold iterated convolution of χ_{r} is of class with support *B*(0, *mr*). Since *r* and *m* are arbitrary, *u* is too. Moreover,

for all 0 < *s* < *r* so that Δ*u* = 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 ∫*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 **R**^{n}, which is less singular at *x*_{0} than the fundamental solution ( for ) , that is

then *f* extends to a harmonic function on Ω (compare Riemann's theorem for functions of a complex variable).

**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).

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

fis bounded, the averages of it over the two balls are arbitrarily close, and sofassumes the same value at any two points.

The proof can be adapted to the case where the harmonic function *f* is merely bounded above or below. By adding a constant and possibly multiplying by , we may assume that *f* is non-negative. Then for any two points and , and any positive number , we let . We then consider the balls and , 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

(Note that since is independent of , we denote it merely as .) In the last expression, we may multiply and divide by and use the averaging property again, to obtain

But as , the quantity

tends to 1. Thus, . The same argument with the roles of and reversed shows that , so that .

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 *H*^{1}(Ω) as the minimizers of the Dirichlet energy integral

with respect to local variations, that is, all functions such that *J*(*u*) ≤ *J*(*u* + *v*) 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 *C*^{2} function that satisfies Δ*f* ≥ 0 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). This kind of harmonic map appears 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 critical 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*^{−1}*TN*.

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.

- ↑ Axler, Sheldon; Bourdon, Paul; Ramey, Wade (2001).
*Harmonic Function Theory*. New York: Springer. p. 25. ISBN 0-387-95218-7. - ↑ Nelson, Edward (1961). "A proof of Liouville's theorem".
*Proceedings of the AMS*.**12**: 995. doi: 10.1090/S0002-9939-1961-0259149-4 .

In the field of complex analysis in mathematics, the **Cauchy–Riemann equations**, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy (1814) then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.

In mathematics, the **Laplace transform**, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

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In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇^{2} or Δ. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ*f*(*p*) of a function *f* at a point *p* measures by how much the average value of *f* over small spheres or balls centered at *p* deviates from *f*(*p*).

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In potential theory, the **Poisson kernel** is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

In mathematics, **Harnack's inequality** is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). J. Serrin (1955), and J. Moser generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by R. Hamilton (1993), for the Ricci flow. Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. Harnack's inequality can also be used to show the interior regularity of weak solutions of partial differential equations.

In mathematics, **subharmonic** and **superharmonic** functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

In mathematics, a **locally integrable function** is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to *L*^{p} spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain : in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.

A **parabolic partial differential equation** is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments.

A **linear response function** describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.

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In mathematics, **differential forms on a Riemann surface** are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1940). This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.

In numerical mathematics, the **gradient discretisation method** (**GDM**) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes.

- Evans, Lawrence C. (1998),
*Partial Differential Equations*, American Mathematical Society. - Gilbarg, David; Trudinger, Neil,
*Elliptic Partial Differential Equations of Second Order*, ISBN 3-540-41160-7 . - Han, Q.; Lin, F. (2000),
*Elliptic Partial Differential Equations*, American Mathematical Society. - Jost, Jürgen (2005),
*Riemannian Geometry and Geometric Analysis*(4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7 .

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