Editing Harmonic function (section)

=== The mean value property ===
If {{math|''B''(''x'', ''r'')}} is a [[Ball (mathematics)|ball]] with center {{mvar|x}} and radius {{mvar|r}} which is completely contained in the open set <math>\Omega \subset \R^n,</math> then the value {{math|''u''(''x'')}} of  a harmonic function <math>u: \Omega \to \R</math> at the center of the ball is given by the average value of {{mvar|u}} on the surface of the ball; this average value is also equal to the average value of {{mvar|u}}  in the interior of the ball.  In other words,
<math display="block">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>
where {{mvar|ω{{sub|n}}}} is the volume of the unit ball in {{mvar|n}} dimensions and {{mvar|σ}} is the {{math|(''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 [[convolution]]s, if
<math display="block">\chi_r := \frac{1}{|B(0, r)|}\chi_{B(0, r)} = \frac{n}{\omega_n r^n}\chi_{B(0, r)}</math>
denotes the [[indicator function|characteristic function]] of the ball with radius {{mvar|r}} about the origin, normalized so that <math display="inline">\int_{\R^n}\chi_r\, dx = 1,</math> the function {{mvar|u}} is harmonic on {{math|Ω}} if and only if
<math display="block">u(x) = u*\chi_r(x)\;</math>
for all x and r such that <math>B(x,r) \subset \Omega.</math>

'''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 {{math|0 < ''s'' < ''r''}}
<math display="block">\Delta w = \chi_r  - \chi_s\;</math>
admits an easy explicit solution {{mvar|w{{sub|r,s}}}} of class {{math|''C''<sup>1,1</sup>}} with compact support in {{math|''B''(0, ''r'')}}. Thus, if {{mvar|u}} is harmonic in {{math|Ω}}
<math display="block">0=\Delta u * w_{r,s} = u*\Delta w_{r,s}= u*\chi_r  - u*\chi_s\;</math>
holds in the set {{math|Ω{{sub|''r''}}}} of all points {{mvar|x}} in {{math|Ω}} with <math>\operatorname{dist}(x,\partial\Omega) > r.</math>

Since {{mvar|u}} is continuous in {{math|Ω}}, <math>u * \chi_s</math> converges to {{mvar|u}} as {{math|''s'' → 0}} showing the mean value property for {{mvar|u}} in {{math|Ω}}. Conversely, if {{mvar|u}} is any <math>L^1_{\mathrm{loc}}\;</math> function satisfying the mean-value property in {{math|Ω}}, that is,
<math display="block">u*\chi_r = u*\chi_s\;</math>
holds in {{math|Ω{{sub|''r''}}}} for all {{math|0 < ''s'' < ''r''}} then, iterating {{mvar|m}} times the convolution with {{math|χ{{sub|''r''}}}} one has:
<math display="block">u = u*\chi_r = u*\chi_r*\cdots*\chi_r\,,\qquad x\in\Omega_{mr},</math>
so that {{mvar|u}} is <math>C^{m-1}(\Omega_{mr})\;</math> because the {{mvar|m}}-fold iterated convolution of {{math|χ{{sub|''r''}}}} is of class <math>C^{m-1}\;</math> with support  {{math|''B''(0, ''mr'')}}. Since {{mvar|r}} and {{mvar|m}} are arbitrary, {{mvar|u}} is <math>C^{\infty}(\Omega)\;</math> too. Moreover,
<math display="block">\Delta u * w_{r,s} = u*\Delta w_{r,s} = u*\chi_r  - u*\chi_s = 0\;</math>
for all {{math|0 < ''s'' < ''r''}} so that {{math|1=Δ''u'' = 0}} in {{math|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 {{mvar|h}} is any spherically symmetric function [[Support (mathematics)|supported]] in {{math|''B''(''x'', ''r'')}} such that <math display="inline">\int h = 1,</math> then <math>u(x) = h * u(x).</math> In other words, we can take the weighted average of {{mvar|u}} about a point and recover {{math|''u''(''x'')}}. In particular, by taking {{mvar|h}} to be a {{math|''C''<sup>∞</sup>}} function, we can recover the value of {{mvar|u}} at any point even if we only know how {{mvar|u}} acts as a [[Distribution (mathematics)|distribution]]. See [[Weyl's lemma (Laplace equation)|Weyl's lemma]].