Editing Harmonic function (section)

== Properties of harmonic functions ==
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 [[analytic function|real analytic]].

=== Maximum principle ===
Harmonic functions satisfy the following ''[[maximum modulus principle|maximum principle]]'': if {{mvar|K}} is a nonempty [[Compact space|compact subset]] of {{mvar|U}}, then {{mvar|f }} restricted to {{mvar|K}} attains its [[maxima and minima|maximum and minimum]] on the [[boundary (topology)|boundary]] of {{mvar|K}}.  If {{mvar|U}} is [[connected space|connected]], this means that {{mvar|f }} cannot have local maxima or minima, other than the exceptional case where {{mvar|f }} is [[constant function|constant]]. Similar properties can be shown for [[subharmonic function]]s.

=== 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]].

=== Harnack's inequality ===
Let 
<math display="block">V \subset \overline{V} \subset \Omega</math>
be a  connected set in a bounded domain {{math|Ω}}.
Then for every non-negative harmonic function {{mvar|u}},
[[Harnack's inequality]] 
<math display="block">\sup_V u \le C \inf_V u</math>
holds for some constant {{mvar|C}} that depends only on {{mvar|V}} and {{math|Ω}}.

=== Removal of singularities ===
The following principle of removal of singularities holds for harmonic functions. If {{mvar|f}} is a harmonic function defined on a dotted open subset <math>\Omega \smallsetminus \{x_0\}</math> of {{tmath|\R^n}}, which is less singular at {{math|''x''{{sub|0}}}} than the fundamental solution (for {{math|''n'' > 2}}), that is
<math display="block">f(x)=o\left( \vert x-x_0 \vert^{2-n}\right),\qquad\text{as }x\to x_0,</math>
then {{mvar|f}} extends to a harmonic function on {{math|Ω}} (compare [[removable singularity#Riemann's theorem|Riemann's theorem]] for functions of a complex variable).

=== Liouville's theorem ===
'''Theorem''': If {{mvar|f}} is a harmonic function defined on all of {{tmath|\R^n}} which is bounded above or bounded below, then {{mvar|f}} is constant.

(Compare [[Liouville's theorem (complex analysis)|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,<ref>{{cite journal |first=Edward |last=Nelson |title=A proof of Liouville's theorem |journal=[[Proceedings of the American Mathematical Society]] |year=1961 |volume=12 |issue=6 |pages=995 |doi=10.1090/S0002-9939-1961-0259149-4 |doi-access=free }}</ref> using the mean value property mentioned above: 
<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 {{mvar|f }} is bounded, the averages of it over the two balls are arbitrarily close, and so {{mvar|f }} assumes the same value at any two points.  </blockquote>
The proof can be adapted to the case where the harmonic function {{mvar|f }} is merely bounded above or below. By adding a constant and possibly multiplying by –1, we may assume that {{mvar|f }} is non-negative. Then for any two points {{mvar|x}} and {{mvar|y}}, and any positive number {{mvar|R}}, we let <math>r=R+d(x,y).</math> We then consider the balls {{math|''B{{sub|R}}''(''x'')}} and {{math|''B{{sub|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
<math display="block">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.</math>
(Note that since {{math|vol ''B{{sub|R}}''(''x'')}} is independent of {{mvar|x}}, we denote it merely as {{math|vol ''B{{sub|R}}''}}.) In the last expression, we may multiply and divide by {{math|vol ''B{{sub|r}}''}} and use the averaging property again, to obtain
<math display="block">f(x)\leq \frac{\operatorname{vol}(B_r)}{\operatorname{vol}(B_R)}f(y).</math>
But as <math>R\rightarrow\infty ,</math> the quantity 
<math display="block">\frac{\operatorname{vol}(B_r)}{\operatorname{vol}(B_R)} = \frac{\left(R+d(x,y)\right)^n}{R^n}</math> 
tends to 1. Thus, <math>f(x)\leq f(y).</math> The same argument with the roles of {{mvar|x}} and {{mvar|y}} reversed shows that <math>f(y)\leq f(x)</math>, so that <math>f(x) = f(y).</math>

Another proof uses the fact that given a [[Wiener process|Brownian motion]] {{mvar|B{{sub|t}}}} in {{tmath|\R^n,}} such that <math>B_0 = x_0,</math> we have <math>E[f(B_t)] = f(x_0)</math> for all {{math|''t'' ≥ 0}}. In words, it says that a harmonic function defines a [[Martingale (probability theory)|martingale]] for the Brownian motion. Then a [[Coupling (probability)|probabilistic coupling]] argument finishes the proof.<ref>{{Cite web |date=2012-01-24 |title=Probabilistic Coupling |url=https://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/ |archive-url=https://web.archive.org/web/20210508091536/https://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/ |archive-date=8 May 2021 |access-date=2022-05-26 |website=Blame It On The Analyst |language=en}}</ref>