A real-valued function , defined in a domain of a Euclidean space , , having continuous partial derivatives of the first and second orders in , and which is a solution of the Laplace equation
where are the orthogonal Cartesian coordinates of the point . This definition is sometimes extended to include complex functions as well, in the sense that their real and imaginary parts and are harmonic functions. The requirements of continuity and even of the existence of derivatives are not a priori indispensable. For instance, one of Privalov's theorems is applicable: A continuous function in is a harmonic function if and only if at any point the mean-value property
— where is the ball of radius with centre at , is the volume of this ball and is the volume element in — is fulfilled for sufficiently small .
If is unbounded with a compact boundary , the definition of a harmonic function may be completed to include the point at infinity , i.e. it may additionally be defined in domains in the Aleksandrov compactification of . The general principle of such a completion of the definition is that, under the simplest transformations which preserve harmonicity (inversion if , Kelvin transformations if , cf. Kelvin transformation) and map a finite point into , a harmonic function in a neighbourhood of becomes a harmonic function in a neighbourhood of . On this basis, a harmonic function is said to be regular at infinity for if
Thus, for a harmonic function which is regular at infinity one always has if . If , the condition
which implies the existence of a finite limit
must be met. Harmonic functions in unbounded domains are usually understood to mean harmonic functions regular at infinity.
In the theory of harmonic functions an important role is played by the principal fundamental solutions of the Laplace equation:
where is the surface area of the unit sphere in . If , this is a harmonic function. The fundamental solutions may be used to write down the basic formula of the theory of harmonic functions, which expresses the values of a harmonic function inside a domain in terms of its values on the boundary and the values of its derivative in the direction of the exterior normal towards at the point :
This Green formula is valid, for example, if the function and its partial derivatives of the first order are continuous in the closed domain , i.e. if , the boundary of which is a piecewise-smooth closed surface or curve. It yields a representation of an arbitrary harmonic function as the sum of single- and double-layer potentials (cf. Potential theory). The densities of these potentials, i.e. the boundary values and respectively, cannot be specified arbitrarily. There is an integral relationship between the two, in that the left-hand side of the last-named formula — the Green integral — must vanish for all points outside . The basic formula of the theory of harmonic functions is a direct analogue of the fundamental formula of the theory of analytic functions — the integral formula of Cauchy (cf. Cauchy integral). This formula also remains valid if the principal fundamental solution in it is replaced by any other fundamental solution of the Laplace equation which is sufficiently smooth in , e.g. belongs to .
The fundamental properties of harmonic functions, on the assumption that the boundary of the domain is piecewise smooth, are listed below. After suitable modification, many of them are also valid for complex harmonic functions.
1) If is a bounded domain and a harmonic function , then
2) The mean-value theorem: If is a harmonic function in the ball of radius with centre at and if , then its value at the centre of the ball is equal to the value of its arithmetical mean on the sphere , i.e.
where is the surface area of the sphere of radius in . If is only continuous, this property may be taken as the definition of a harmonic function.
3) The maximum/minimum principle: Let be a domain in not containing as an interior point. If is a harmonic function in , and , then cannot attain a local extremum at any point , i.e. in any neighbourhood of any point there exists a point at which and there exists a point at which (the maximum/minimum principle in local form). If, in addition, , then the largest and the least values of on are attained only at the points of the boundary (the maximum/minimum principle in global form). Consequently, if on , then everywhere in .
This principle may be generalized in various ways.
For instance, if is a harmonic function in a domain not containing and if
for all points (boundary in ), then everywhere in .
4) The theorem on removable singularities: If is a harmonic function in a domain , , which satisfies the condition
then there exists a finite limit
and completed by the value is a harmonic function in .
5) If is a harmonic function throughout the space , , bounded from above or from below, then .
6) If is a harmonic function in a neighbourhood of a point , then can be expanded in this neighbourhood into a power series in the variables , i.e. all harmonic functions are analytic functions of the variables ; consequently, a harmonic function has derivatives of all orders:
which are also harmonic functions.
7) The uniqueness property: If is a harmonic function in a domain and in some -dimensional neighbourhood of some point , then in . If is an analytic function of the real variables in a domain and if is a harmonic function in some -dimensional neighbourhood of an arbitrary point , then is a harmonic function in .
8) The symmetry principle: Let the boundary of a domain contain a set that is open in the plane , let be a harmonic function in such that and continuous on and let be the domain symmetrical to with respect to the plane ; then can be harmonically extended into the domain by the formula
9) Harnack's first theorem: If a sequence of harmonic functions in a bounded domain , continuous in the closed domain , converges uniformly on the boundary , then it converges uniformly on , and the limit function
is a harmonic function in .
10) Harnack's second theorem: If a sequence of harmonic functions is monotone in a domain and converges at least at one point , then it converges everywhere in towards a harmonic function
There exists a close connection between harmonic functions of two variables and analytic functions of the complex variable . The real and the imaginary part of an analytic function are, possibly multi-valued, conjugate harmonic functions, i.e. they are connected by the Cauchy–Riemann conditions. If a harmonic function is defined in a neighbourhood of a point , the simplest way of finding an analytic function , , for which is given by the Goursat formula:
where and is an arbitrary real constant. Certain spatial problems in mathematical physics also involve multi-valued harmonic functions in domains in , .
The major importance of harmonic functions in mathematical physics is mainly due to the frequent occurrence of vector fields of the form . Such fields in domains not containing field sources must satisfy the conservation equation , i.e. the Laplace equation, which means that in such domains is a harmonic function.
Examples. If is the force vector of the gravity field, is the Newton potential of the gravitational forces; if is the field of velocities of a stationary motion of an incompressible homogeneous gas or liquid, is the velocity potential; if is the strength of an electrostatic field in a homogeneous isotropic medium, is the potential of the electrostatic field; if is the strength of a stationary magnetic field in a homogeneous isotropic medium, is the scalar, usually multi-valued, potential of the magnetic field. In the case of steady propagation of heat in a uniform isotropic medium or a stationary distribution of diffusing particles, the harmonic function is the temperature of the medium or the density of the particles at a point , respectively. Many important problems in the theory of elasticity and in the theory of electromagnetic fields can also be reduced to solving problems concerning harmonic functions.
The boundary Dirichlet problem, or the first boundary value problem, is of special importance in the development of the theory of harmonic functions and mathematical physics. It consists in finding a function which is harmonic in a domain and continuous on , from given continuous values on the boundary . If the surface or line is sufficiently smooth, the solution may be expressed by the Green function :
In the case of the simplest domains (spheres, half-spaces), when the normal derivative is readily expressed in explicit form, the Poisson integral is obtained. The second boundary value problem, or the Neumann problem, is also often encountered. It consists in determining a harmonic function from given values of its normal derivative on the boundary . This problem can be solved using the corresponding Green function, but explicit expressions are much more complicated in this case. There are many more boundary value problems in the theory of harmonic functions, the formulations and solutions of which are more complicated. See also Balayage method; Robin problem.
A special place in the modern theory of harmonic functions is occupied by ill-posed problems, mainly those connected with the Cauchy problem for the Laplace equation. These include, for example, the following problem on the best majorant: If on the boundary of a domain the function and the conditions , are given, find the best possible estimate of in the class of harmonic functions in , .
The study of the boundary properties of harmonic functions related with subharmonic functions (cf. Subharmonic function) and with the boundary properties of analytic functions is of importance. For instance, a harmonic function in the unit ball of usually has no radial limit values
However, in the case of the class of harmonic functions defined by the condition
where is the surface element of , , the radial boundary values exist almost-everywhere on with respect to Lebesgue measure, and an can be represented in the form of a Poisson–Stieltjes integral
is the Poisson kernel and is the Borel measure on . The proper subclass of the class consisting of all harmonic functions that can be represented in by a Poisson–Lebesgue integral,
is also of importance.
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Fundamental property 5) above is also called Picard's theorem (for harmonic functions).
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Harmonic function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Harmonic_function&oldid=28211