|Mathematical analysis → Complex analysis|
|Geometric function theory|
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is,
everywhere on U. This is usually written as
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.
Examples of harmonic functions of two variables are:
Examples of harmonic functions of three variables are given in the table below with
|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|
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 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 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 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 (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function u on an open subset Ω of is locally the real part of a holomorphic function. This is immediately seen observing that, writing the complex function is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, g locally has a primitive f, and u is the real part of f up to a constant, as ux is the real part of
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 then the value u(x) of a harmonic function 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 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.
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
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 wr,s of class C1,1 with compact support in B(0, r). Thus, if u is harmonic in Ω
holds in the set Ωr of all points x in Ω with
Since u is continuous in Ω, 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 Ωr for all 0 < s < r then, iterating m times the convolution with χr one has:
so that u is because the m-fold iterated convolution of χr 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 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 then 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 , which is less singular at x0 than the fundamental solution (for n > 2), 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 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,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.
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 –1, we may assume that f is non-negative. Then for any two points x and y, and any positive number R, we let We then consider the balls BR(x) and BR(y) 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 vol BR(x) is independent of x, we denote it merely as vol BR.) In the last expression, we may multiply and divide by vol Br and use the averaging property again, to obtain
But as the quantity
tends to 1. Thus, The same argument with the roles of x and y reversed shows that , so that
Another proof uses the fact that given a Brownian motion Bt in such that we have for all t ≥ 0. In words, it says that a harmonic function defines a martingale for the Brownian motion. Then a probabilistic coupling argument finishes the proof.
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 Δf ≥ 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 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 is defined to be a critical point of the Dirichlet energy
in which 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
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.
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 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 then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
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 scalar function on Euclidean space. It is usually denoted by the symbols , , or . 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 Δf (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).
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.
In mathematics, a real-valued function defined on a connected open set is said to have a conjugate (function) if and only if they are respectively the real and imaginary parts of a holomorphic function of the complex variable That is, is conjugate to if is holomorphic on As a first consequence of the definition, they are both harmonic real-valued functions on . Moreover, the conjugate of if it exists, is unique up to an additive constant. Also, is conjugate to if and only if is conjugate to .
Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms, sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. J. Serrin (1955), and J. Moser generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions.
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature, such that an initial unit of heat energy is placed at a point at time t = 0.
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 Lp 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.
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In mathematics, a Caccioppoli set is a set whose boundary is measurable and has a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation.
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem.
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In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where is allowed to range over . It is written as
In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained.
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