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| {{about|harmonic functions in mathematics|harmonic function in music|diatonic functionality}}
| | My name is Adela Hannan. I life in Columbia (United States).<br><br>Look into my webpage ... [http://etudesu.org Blog] |
| [[Image:Laplace's equation on an annulus.jpg|right|thumb|300px|A harmonic function defined on an [[Annulus (mathematics)|annulus]].]]
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| In [[mathematics]], [[mathematical physics]] and the theory of [[stochastic process]]es, a '''harmonic function''' is a twice [[derivative|continuously differentiable]] [[function (mathematics)|function]] ''f'' : ''U'' → '''R''' (where ''U'' is an [[open set|open subset]] of '''R'''<sup>''n''</sup>) which satisfies [[Laplace's equation]], i.e.
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| :<math> \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math>
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| everywhere on ''U''. This is usually written as
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| :<math> \nabla^2 f = 0 </math>
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| or
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| :<math>\textstyle \Delta f = 0</math>
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| == Examples ==
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| Examples of harmonic functions of two variables are:
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| * The real and imaginary part of any [[holomorphic function]]
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| * The function <math>\,\! f(x,y)=e^{x} \sin y</math>, this is a special case of the example above, as <math>f(x,y)=\text{Im}(e^{x+iy})</math>, and <math>e^{x+iy}</math> is a [[holomorphic function]].
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| * The function
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| ::<math>\,\! f(x_1,x_2)=\ln ({x_1}^2+{x_2}^2)</math>
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| : defined on <math>\mathbb{R}^2 \backslash \lbrace 0 \rbrace</math> (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass)
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| Examples of harmonic functions of three variables are given in the table below with <math>r^2=x^2+y^2+z^2</math>:
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| :{| class="wikitable"
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| ! Function !! [[Mathematical singularity|Singularity]]
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| |-
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| |align=center|<math>\frac{1}{r}</math>
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| |Unit point charge at origin
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| |-
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| |align=center|<math>\frac{x}{r^3}</math>
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| |x-directed dipole at origin
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| |-
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| |align=center|<math>-\ln(r^2-z^2)\,</math>
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| |Line of unit charge density on entire z-axis
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| |-
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| |align=center|<math>-\ln(r+z)\,</math>
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| |Line of unit charge density on negative z-axis
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| |-
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| |align=center|<math>\frac{x}{r^2-z^2}\,</math>
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| |Line of x-directed dipoles on entire z axis
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| |-
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| |align=center|<math>\frac{x}{r(r+z)}\,</math>
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| |Line of x-directed dipoles on negative z axis
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| |}
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| Harmonic functions that arise in physics are determined by their [[mathematical singularity|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 goes to 0 as you go to infinity. In this case, uniqueness follows by [[Liouville's theorem (complex analysis)|Liouville's theorem]].
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| The singular points of the harmonic functions above are expressed as "[[Charge (physics)|charges]]" and "[[Charge density|charge densities]]" using the terminology of [[electrostatics]], and so the corresponding harmonic function will be proportional to the [[Electric potential|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 [[Method of inversion|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.
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| Finally, examples of harmonic functions of ''n'' variables are:
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| * The constant, linear and affine functions on all of '''R'''<sup>''n''</sup> (for example, the [[electric potential]] between the plates of a [[capacitor]], and the [[gravity potential]] of a slab)
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| * The function <math>\,\! f(x_1,\dots,x_n)=({x_1}^2+\cdots+{x_n}^2)^{1-n/2}</math> on <math>\mathbb{R}^n \backslash \lbrace 0 \rbrace</math> for ''n'' > 2.
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| == Remarks ==
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| The set of harmonic functions on a given open set ''U'' can be seen as the [[Kernel (linear operator)|kernel]] of the [[Laplace operator]] Δ and is therefore a [[vector space]] over '''R''': sums, differences and scalar multiples of harmonic functions are again harmonic.
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| If ''f'' is a harmonic function on ''U'', then all [[partial derivative]]s of ''f'' are also harmonic functions on ''U''. The Laplace operator Δ and the partial derivative operator will commute on this class of functions.
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| In several ways, the harmonic functions are real analogues to [[holomorphic function]]s. All harmonic functions are [[analytic function|analytic]], i.e. they can be locally expressed as [[power series]]. This is a general fact about [[elliptic operator]]s, of which the Laplacian is a major example.
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| The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because any continuous function satisfying the mean value property is harmonic. Consider the sequence on (−∞, 0) × '''R''' defined by <math>\scriptstyle f_n(x,y) = \frac1n \exp(nx)\cos(ny)</math>. 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.
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| == Connections with complex function theory ==
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| The real and imaginary part of any holomorphic function yield harmonic functions on '''R'''<sup>2</sup> (these are said to be a pair of [[harmonic conjugate]] functions). Conversely, any harmonic function ''u'' on an open subset Ω of '''R'''<sup>2</sup> 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<sub>x</sub>'' − i ''u<sub>y</sub>'' is holomorphic in Ω because it satisfies the [[Cauchy–Riemann equations]]. Therefore, ''g'' has locally a primitive ''f'', and ''u'' is the real part of ''f'' up to a constant, as ''u<sub>x</sub>'' is the real part of <math>\scriptstyle f\,^\prime=g</math> .
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| Although the above correspondence with holomorphic functions only holds for functions of two real variables, still harmonic functions in ''n'' variables 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 one holds for them in analogy to the corresponding theorems in complex functions theory.
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| == Properties of harmonic functions ==
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| Some important properties of harmonic functions can be deduced from Laplace's equation.
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| === Regularity theorem for harmonic functions ===
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| Harmonic functions are infinitely differentiable. In fact, harmonic functions are [[analytic function|real analytic]].
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| === Maximum principle ===
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| Harmonic functions satisfy the following ''[[maximum principle]]'': if ''K'' is any [[Compact space|compact subset]] of ''U'', then ''f'', restricted to ''K'', attains its [[maxima and minima|maximum and minimum]] on the [[boundary (topology)|boundary]] of ''K'', unless that minima is 0. If ''U'' is [[connected space|connected]], this means that ''f'' cannot have non-zero local maxima or minima, other than the exceptional case where ''f'' is [[constant function|constant]]. Similar properties can be shown for subharmonic functions.
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| === The mean value property ===
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| If ''B''(''x'', ''r'') is a [[Ball (mathematics)|ball]] with center ''x'' and radius ''r'' which is completely contained in the open set Ω ⊂ '''R'''<sup>''n''</sup>, 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
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| :<math> 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</math>
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| where ω<sub>''n''</sub> is the volume of the [[unit ball]] in ''n'' dimensions and σ is the ''n''-1 dimensional surface measure .
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| Conversely, all locally integrable functions satisfying the (volume) mean-value property are both infinitely differentiable and harmonic.
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| In terms of [[convolution]]s, if
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|
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| :<math>\chi_r:=\frac{1}{|B(0,r)|}\chi_{B(0,r)}=\frac{1}{\omega_n r^n}\chi_{B(0,r)}</math>
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| denotes the [[indicator function|characteristic function]] of the ball with radius ''r'' about the origin, normalized so that <math>\scriptstyle \int_{\mathbf{R}^n}\chi_r\, dx=1</math>, the function ''u'' is harmonic on Ω if and only if
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| :<math>u(x) = u*\chi_r(x)\;</math>
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| as soon as ''B''(''x'', ''r'') ⊂ Ω.
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| '''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''
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| :<math>\Delta w = \chi_r - \chi_s\;</math>
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| admits an easy explicit solution ''w<sub>r,s</sub>'' of class ''C''<sup>1,1</sup> with compact support in ''B''(0, ''r''). Thus, if ''u'' is harmonic in Ω
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| :<math>0=\Delta u * w_{r,s} = u*\Delta w_{r,s}= u*\chi_r - u*\chi_s\;</math>
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| holds in the set Ω<sub>''r''</sub> of all points ''x'' in '''R'''<sup>''n''</sup> with <math>\mathrm{dist}(x,\partial\Omega)<r</math> .
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| Since ''u'' is continuous in Ω, ''u''*χ<sub>''r''</sub> converges to ''u'' as ''s'' → 0 showing the mean value property for ''u'' in Ω. Conversely, if ''u'' is any <math>L^1_{\mathrm{loc}}\;</math> function satisfying the mean-value property in Ω, that is,
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| :<math>u*\chi_r = u*\chi_s\;</math>
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| holds in Ω<sub>''r''</sub> for all 0 < ''s'' < ''r'' then, iterating ''m'' times the convolution with χ<sub>''r''</sub> one has:
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| :<math>u = u*\chi_r = u*\chi_r*\cdots*\chi_r\,,\qquad x\in\Omega_{mr},</math>
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| so that ''u'' is <math>C^{m-1}(\Omega_{mr})\;</math> because the m-fold iterated convolution of χ<sub>''r''</sub> is of class <math>C^{m-1}\;</math> with support ''B''(0, ''mr''). Since ''r'' and ''m'' are arbitrary, ''u'' is <math>C^{\infty}(\Omega)\;</math> too. Moreover
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| <math>\Delta u * w_{r,s} = u*\Delta w_{r,s} = u*\chi_r - u*\chi_s=0\;</math>
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| 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.
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| This statement of the mean value property can be generalized as follows: If ''h'' is any spherically symmetric function [[Support (mathematics)|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''<sup>∞</sup> function, we can recover the value of ''u'' at any point even if we only know how ''u'' acts as a [[Distribution (mathematics)|distribution]]. See [[Weyl's lemma (Laplace equation)|Weyl's lemma]].
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| === Harnack's inequality ===
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| Let ''u'' be a non-negative harmonic function in a bounded domain Ω. Then for every connected set
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| :<math>V \subset \overline{V} \subset \Omega,</math>
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| [[Harnack's inequality]]
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| :<math>\sup_V u \le C \inf_V u</math>
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| holds for some constant ''C'' that depends only on ''V'' and Ω.
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| === Removal of singularities ===
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| The following principle of removal of singularities holds for harmonic functions. If ''f'' is a harmonic function defined on a dotted open subset <math> \scriptstyle\Omega\setminus\{x_0\}</math> of '''R'''<sup>''n'' </sup>, which is less singular at ''x''<sub>0</sub> than the fundamental solution, that is
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| :<math>f(x)=o\left( \vert x-x_0 \vert^{2-n}\right),\qquad\mathrm{as\ }x\to x_0,</math>
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| then ''f'' extends to a harmonic function on Ω (compare [[removable singularity#Riemann's theorem|Riemann's theorem]] for functions of a complex variable).
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| === Liouville's theorem ===
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| If ''f'' is a harmonic function defined on all of '''R'''<sup>''n''</sup> which is bounded above or bounded below, then ''f'' is constant (compare [[Liouville's theorem (complex analysis)|Liouville's theorem for functions of a complex variable]]).
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| [[Edward Nelson]] gave a particularly short proof <ref>Edward Nelson, A proof of Liouville's theorem. Proceedings of the AMS, 1961. [http://www.jstor.org/stable/2034412 pdf at JSTOR]</ref> of this theorem, using the mean value property mentioned above:
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| <blockquote>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. </blockquote>
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| ==Generalizations==
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| ===Weakly harmonic function===
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| A function (or, more generally, a [[distribution (mathematics)|distribution]]) is [[weakly harmonic]] if it satisfies Laplace's equation
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| :<math>\Delta f = 0\,</math>
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| in a [[weak derivative|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 (Laplace equation)|Weyl's lemma]].
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| There are other [[weak formulation]]s of Laplace's equation that are often useful. One of which is [[Dirichlet's principle]], representing harmonic functions in the [[Sobolev space]] ''H''<sup>1</sup>(Ω) as the minimizers of the [[Dirichlet energy]] integral
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| :<math>J(u):=\int_\Omega |\nabla u|^2\, dx</math>
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| with respect to local variations, that is, all functions <math>u\in H^1(\Omega)</math> such that ''J''(''u'') ≤ ''J''(''u'' + ''v'') holds for all <math>v\in C^\infty_c(\Omega),</math> or equivalently, for all <math>v\in H^1_0(\Omega).</math>
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| ===Harmonic functions on manifolds===
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| 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
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| :<math>\ \Delta f = 0.</math>
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| 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 equation]]s of the second order.
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| ===Subharmonic functions===
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| A ''C''<sup>2</sup> 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.
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| ===Harmonic forms===
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| One generalization of the study of harmonic functions is the study of [[harmonic form]]s on [[Riemannian manifold]]s, 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]]). These kind of harmonic maps appear 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]].
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| ===Harmonic maps between manifolds===
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| {{main|Harmonic map}}
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| If ''M'' and ''N'' are two Riemannian manifolds, then a [[harmonic map]] {{nowrap|''u'' : ''M'' → ''N''}} is defined to be a stationary point of the Dirichlet energy
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| :<math>D[u] = \frac{1}{2}\int_M \|du\|^2\,d\operatorname{Vol}</math>
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| in which {{nowrap|''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''<sup>−1</sup> ''TN''.
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| Important special cases of harmonic maps between manifolds include [[minimal surface]]s, 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.
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| ==See also==
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| *[[Dirichlet problem]]
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| *[[Dirichlet principle]]
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| *[[Dirichlet energy]]
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| *[[Heat equation]]
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| *[[Laplace's equation]]
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| *[[Poisson's equation]]
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| *[[Quadrature domains]]
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| *[[Subharmonic function]]
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| *[[Harmonic map]]
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| *[[Balayage]]
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| == References ==
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| {{Reflist}}
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| * {{citation|first=Lawrence C.|last=Evans|authorlink=Lawrence C. Evans|year=1998|title=Partial Differential Equations|publisher=American Mathematical Society}}.
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| * {{citation|first1=David|last1=Gilbarg|authorlink1=David Gilbarg|first2=Neil|last2=Trudinger|authorlink2=Neil Trudinger|title=Elliptic Partial Differential Equations of Second Order|isbn=3-540-41160-7}}.
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| * {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=4th | isbn=978-3-540-25907-7 | year=2005}}.
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| * {{citation|first1=Q.|last1=Han|first2=F.|last2=Lin|year=2000|title=Elliptic Partial Differential Equations|publisher=American Mathematical Society}}.
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| == External links ==
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| * {{springer|title=Harmonic function|id=p/h046470}}
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| * {{MathWorld | urlname=HarmonicFunction | title= Harmonic Function }}
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| * [http://math.fullerton.edu/mathews/c2003/HarmonicFunctionMod.html Harmonic Functions Module by John H. Mathews]
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| * [http://www.axler.net/HFT.html Harmonic Function Theory by S.Axler, Paul Bourdon, and Wade Ramey]
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| {{DEFAULTSORT:Harmonic Function}}
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| [[Category:Harmonic functions| ]]
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