دالة توافقية

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 f : UR, where U is an open subset of Rn, that satisfies Laplace's equation, that is,

everywhere on U. This is usually written as

or

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أصل مصطلح "توافقية harmonic"

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 periodic 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]


أمثلة

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 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

خصائص الدوال التوافقية

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

Regularity theorem for harmonic functions

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

مبدأ القيمة العظمى

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.

خاصية القيمة المتوسطة

If B(x, r) is a ball with center x and radius r which is completely contained in the open set Ω ⊂ Rn, 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 ωn 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 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 with .

Since u is continuous in Ω, ur 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 ∫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.

Harnack's inequality

Let u be a non-negative harmonic function in a bounded domain Ω. Then for every connected set

Harnack's inequality

holds for some constant C that depends only on V and Ω.

Removal of singularities

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 x0 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).

Liouville's theorem

Theorem: 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 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 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 , 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 .

التعميمات

Weakly harmonic function

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 J(u) ≤ J(u + v) holds for all or equivalently, for all


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انظر أيضاً

Notes

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

References

  • 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 .

External links