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

${\displaystyle {\frac {\partial ^{2 f {\partial x_{1 ^{2 +{\frac {\partial ^{2 f {\partial x_{2 ^{2 +\cdots +{\frac {\partial ^{2 f {\partial x_{n ^{2 =0$

everywhere on U. This is usually written as

${\displaystyle \nabla ^{2 f=0$

or

${\displaystyle \textstyle \Delta f=0$

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 -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.

أمثلة

Examples of harmonic functions of two variables are:

• The real and imaginary parts of any holomorphic function
• The function ${\displaystyle f(x,y)=e^x\sin <!--\; \; -->y}$
• The function ${\displaystyle \,\!f(x,y)=\ln(x^{2 +y^{2 )$ defined on ${\displaystyle \mathbb {R ^{2 \setminus \lbrace 0\rbrace$. 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 ${\displaystyle r^{2 =x^{2 +y^{2 +z^{2$:

Function	Singularity
${\displaystyle {\frac {1 {r$	Unit point charge at origin
${\displaystyle {\frac {x {r^{3$	x-directed dipole at origin
${\displaystyle -\ln(r^{2 -z^{2 )\,$	Line of unit charge density on entire z-axis
${\displaystyle -\ln(r+z)\,$	Line of unit charge density on negative z-axis
${\displaystyle {\frac {x {r^{2 -z^{2 \,$	Line of x-directed dipoles on entire z axis
${\displaystyle {\frac {x {r(r+z) \,$	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,

${\displaystyle u(x)={\frac {1 {n\omega _{n r^{n-1 \int _{\partial B(x,r) u\,d\sigma ={\frac {1 {\omega _{n r^{n \int _{B(x,r) u\,dV$

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

${\displaystyle \chi _{r :={\frac {1 {|B(0,r)| \chi _{B(0,r) ={\frac {n {\omega _{n r^{n \chi _{B(0,r)$

denotes the characteristic function of the ball with radius r about the origin, normalized so that ${\displaystyle \scriptstyle \int _{\mathbf {R ^{n \chi _{r \,dx=1$, the function u is harmonic on Ω if and only if

${\displaystyle u(x)=u*\chi _{r (x)\;$

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

${\displaystyle \Delta w=\chi _{r -\chi _{s \;$

admits an easy explicit solution w_r,s of class C^1,1 with compact support in B(0, r). Thus, if u is harmonic in Ω

${\displaystyle 0=\Delta u*w_{r,s =u*\Delta w_{r,s =u*\chi _{r -u*\chi _{s \;$

holds in the set Ω_r of all points x in ${\displaystyle \Omega$ with ${\displaystyle \operatorname {dist (x,\partial \Omega )>r$ .

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 ${\displaystyle L_{\mathrm {loc ^{1 \;$ function satisfying the mean-value property in Ω, that is,

${\displaystyle u*\chi _{r =u*\chi _{s \;$

holds in Ω_r for all 0 < s < r then, iterating m times the convolution with χ_r one has:

${\displaystyle u=u*\chi _{r =u*\chi _{r *\cdots *\chi _{r \,,\qquad x\in \Omega _{mr ,$

so that u is ${\displaystyle C^{m-1 (\Omega _{mr )\;$ because the m-fold iterated convolution of χ_r is of class ${\displaystyle C^{m-1 \;$ with support B(0, mr). Since r and m are arbitrary, u is ${\displaystyle C^{\infty (\Omega )\;$ too. Moreover,

${\displaystyle \Delta u*w_{r,s =u*\Delta w_{r,s =u*\chi _{r -u*\chi _{s =0\;$

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

${\displaystyle V\subset {\overline {V \subset \Omega ,$

Harnack's inequality

${\displaystyle \sup _{V u\leq C\inf _{V u$

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 ${\displaystyle \scriptstyle \Omega \,\setminus \,\{x_{0 \$ of Rⁿ , which is less singular at x₀ than the fundamental solution ( for ${\displaystyle n>2$) , that is

${\displaystyle f(x)=o\left(\vert x-x_{0 \vert ^{2-n \right),\qquad {\text{as x\to x_{0 ,$

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, 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 ${\displaystyle -1$, we may assume that f is non-negative. Then for any two points ${\displaystyle x$ and ${\displaystyle y$, and any positive number ${\displaystyle R$, we let ${\displaystyle r=R+d(x,y)$. We then consider the balls ${\displaystyle B_{R (x)$ and ${\displaystyle B_{r (y)$, 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

${\displaystyle f(x)={\frac {1 {\operatorname {vol (B_{R ) \int _{B_{R (x) f(z)\,dz\leq {\frac {1 {\operatorname {vol (B_{R ) \int _{B_{r (y) f(z)\,dz.$

(Note that since ${\displaystyle \operatorname {vol (B_{R (x))$ is independent of ${\displaystyle x$, we denote it merely as ${\displaystyle \operatorname {vol (B_{R )$.) In the last expression, we may multiply and divide by ${\displaystyle \operatorname {vol (B_{r )$ and use the averaging property again, to obtain

${\displaystyle f(x)\leq {\frac {\operatorname {vol (B_{r ) {\operatorname {vol (B_{R ) f(y).$

But as ${\displaystyle R\rightarrow \infty$, the quantity

${\displaystyle {\frac {\operatorname {vol (B_{r ) {\operatorname {vol (B_{R ) ={\frac {(R+d(x,y))^{n {R^{n$

tends to 1. Thus, ${\displaystyle f(x)\leq f(y)$. The same argument with the roles of ${\displaystyle x$ and ${\displaystyle y$ reversed shows that ${\displaystyle f(y)\leq f(x)$, so that ${\displaystyle f(x)=f(y)$.

التعميمات

Weakly harmonic function

A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation

${\displaystyle \Delta f=0\,$

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

${\displaystyle J(u):=\int _{\Omega |\nabla u|^{2 \,dx$

with respect to local variations, that is, all functions ${\displaystyle u\in H^{1 (\Omega )$ such that J(u) ≤ J(u + v) holds for all ${\displaystyle v\in C_{c ^{\infty (\Omega ),$ or equivalently, for all ${\displaystyle v\in H_{0 ^{1 (\Omega ).$

انظر أيضاً

Notes

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

• Hazewinkel, Michiel, ed. (2001), "Harmonic function", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Eric W. Weisstein, Harmonic Function at MathWorld.
• Harmonic Function Theory by S.Axler, Paul Bourdon, and Wade Ramey