Διαφορά μεταξύ των αναθεωρήσεων του «Αρμονική συνάρτηση»

καμία σύνοψη επεξεργασίας
τότε η f επεκτείνεται σε μια αρμονική συνάρτηση στο Ω. (Σύγκριση με το θεώρημα Riemann για συναρτήσεις μιας μιγαδικής μεταβλητής).
 
=== Θεώρημα του Liouville ===
== Properties of harmonic functions ==
Εάν f είναι μια αρμονική συνάρτηση, άνω και κάτω φραγμένη, ορισμένη σε ολόκληρο το '''R'''<sup>''n''</sup> '',''τότε η f είναι σταθερή. Ο Edward Nelson παρουσίασε μια εξαιρετικά μικρή απόδειξη του θεωρήματος αυτού, βασισμένη στην ιδιότητα της μέσης τιμής που αναφέρθηκε παραπάνω:<blockquote>Δοσμένων δυο σημείων, επιλέξτε δυο μπάλες με κέντρα τα σημεία αυτά και ακτίνες ίσες. Εάν οι ακτίνες είναι αρκετά μεγάλες, τότε οι δυο μπάλες θα συμπίπτουν εκτός από μια αυθαίρετα μικρή περιοχή του όγκου τους. Εφόσον η f είναι φραγμένη, η μέση τιμή της πανω στις δυο μπάλες θα είναι τόσο κοντά, ώστε η f να θεωρείται οτι παίρνει την ίδια τιμή σε κάθε ζεύγος σημείων.</blockquote>
Some important properties of harmonic functions can be deduced from Laplace's equation.
 
=== Regularity theorem for harmonic functions ===
Harmonic functions are infinitely differentiable. In fact, harmonic functions are [[:en:Analytic_function|real analytic]].
 
=== Maximum principle ===
Harmonic functions satisfy the following ''[[:en:Maximum_principle|maximum principle]]'': if ''K'' is any [[:en:Compact_space|compact subset]] of ''U'', then ''f'', restricted to ''K'', attains its [[:en:Maxima_and_minima|maximum and minimum]] on the [[:en:Boundary_(topology)|boundary]] of ''K''. If ''U'' is [[:en:Connected_space|connected]], this means that ''f'' cannot have local maxima or minima, other than the exceptional case where ''f'' is [[:en:Constant_function|constant]]. Similar properties can be shown for subharmonic functions.
 
=== The mean value property ===
If ''B''(''x'', ''r'') is a [[:en: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
: <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>
where ω<sub>''n''</sub> is the volume of the [[:en:Unit_sphere|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 [[:en:Convolution|convolutions]], if
: <math>\chi_r:=\frac{1}{|B(0,r)|}\chi_{B(0,r)}=\frac{1}{\omega_n r^n}\chi_{B(0,r)}</math>
denotes the [[:en: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
: <math>u(x) = u*\chi_r(x)\;</math>
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''
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.
 
This statement of the mean value property can be generalized as follows: If ''h'' is any spherically symmetric function [[:en: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 [[:en:Distribution_(mathematics)|distribution]]. See [[:en:Weyl's_lemma_(Laplace_equation)|Weyl's lemma]].<blockquote></blockquote>
 
=== Harnack's inequality ===
Let ''u'' be a non-negative harmonic function in a bounded domain Ω. Then for every connected set
: <math>V \subset \overline{V} \subset \Omega,</math>
[[:en:Harnack's_inequality|Harnack's inequality]]
: <math>\sup_V u \le C \inf_V u</math>
holds for some constant ''C'' that depends only on ''V'' and Ω.
 
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
: <math>f(x)=o\left( \vert x-x_0 \vert^{2-n}\right),\qquad\text{as }x\to x_0,</math>
then ''f'' extends to a harmonic function on Ω (compare [[:en:Removable_singularity#Riemann's_theorem|Riemann's theorem]] for functions of a complex variable).
 
=== Liouville's theorem ===
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 [[:en:Liouville's_theorem_(complex_analysis)|Liouville's theorem for functions of a complex variable]]).
 
[[:en:Edward_Nelson|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:<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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