Διαφορά μεταξύ των αναθεωρήσεων του «Αρμονική συνάρτηση»

καμία σύνοψη επεξεργασίας
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>
 
== Γενικεύσεις ==
'''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''
: <math>\Delta w = \chi_r - \chi_s\;</math>
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
: <math>0=\Delta u * w_{r,s} = u*\Delta w_{r,s}= u*\chi_r - u*\chi_s\;</math>
holds in the set Ω<sub>''r''</sub> of all points ''x'' in <math> \Omega </math> with <math>\mathrm{dist}(x,\partial\Omega)>r</math> .
 
=== Ασθενής αρμονική συνάρτηση ===
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,
Μια συνάρτηση (ή γενικότερα μια κατανομή) είναι ασθενώς αρμονική εάν ικανοποιεί ασθενώς την εξίσωση του Λαπλάς <math>\Delta f = 0\,</math>. Μια ασθενώς αρμονική συνάρτηση συμπίπτει σχεδόν εξ'ολοκλήρου με μια αρμονική συνάρτηση, και είναι συγκεκριμένα, λεία. Μια ασθενώς αρμονική κατανομή είναι ακριβώς η κατανομή εκείνη που σχετίζεται με μια αρμονική συνάρτηση, είναι όμως επιπλέον και λεία. Αυτό είναι το λήμμα του Γουέιλ.
: <math>u*\chi_r = u*\chi_s\;</math>
holds in Ω<sub>''r''</sub> for all 0 < ''s'' < ''r'' then, iterating ''m'' times the convolution with χ<sub>''r''</sub> one has:
: <math>u = u*\chi_r = u*\chi_r*\cdots*\chi_r\,,\qquad x\in\Omega_{mr},</math>
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
 
== Generalizations ==
<math>\Delta u * w_{r,s} = u*\Delta w_{r,s} = u*\chi_r - u*\chi_s=0\;</math>
 
=== Weakly harmonic function ===
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.
A function (or, more generally, a [[:en:Distribution_(mathematics)|distribution]]) is [[:en:Weakly_harmonic|weakly harmonic]] if it satisfies Laplace's equation
: <math>\Delta wf = 0\chi_r - \chi_s\;,</math>
in a [[:en: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 [[:en:Weyl's_lemma_(Laplace_equation)|Weyl's lemma]].
 
There are other [[:en:Weak_formulation|weak formulations]] of Laplace's equation that are often useful. One of which is [[:en:Dirichlet's_principle|Dirichlet's principle]], representing harmonic functions in the [[:en:Sobolev_space|Sobolev space]] ''H''<sup>1</sup>(Ω) as the minimizers of the [[:en:Dirichlet_energy|Dirichlet energy]] integral
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>
: <math>J(u):=\int_\Omega |\nabla u|^2\, dx</math>
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>
 
=== Harmonic functions on manifolds ===
Harmonic functions can be defined on an arbitrary [[:en:Riemannian_manifold|Riemannian manifold]], using the [[:en:Laplace–Beltrami_operator|Laplace–Beltrami operator]] Δ. In this context, a function is called ''harmonic'' if
: <math>u*\chi_r \Delta f = u*\chi_s\;0.</math>
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 [[:en:Geodesic|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 [[:en:Elliptic_partial_differential_equation|elliptic partial differential equations]] of the second order.
 
=== Subharmonic functions ===
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.
 
=== Harmonic forms ===
One generalization of the study of harmonic functions is the study of [[:en:Harmonic_form|harmonic forms]] on [[:en:Riemannian_manifold|Riemannian manifolds]], and it is related to the study of [[:en:Cohomology|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 [[:en:Dirichlet_principle|Dirichlet principle]]). This 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 [[:en:Geodesic|geodesic]].
 
=== Harmonic maps between manifolds ===
{{main|Harmonic map}}If ''M'' and ''N'' are two Riemannian manifolds, then a [[:en:Harmonic_map|harmonic map]] {{nowrap|''u'' : ''M'' &rarr; ''N''}} is defined to be a critical point of the Dirichlet energy
: <math>D[u] = \frac{1}{2}\int_M \|du\|^2\,d\operatorname{Vol}</math>
in which {{nowrap|''du'' : ''TM'' &rarr; ''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>&#x2212;1</sup> ''TN''.
 
Important special cases of harmonic maps between manifolds include [[:en:Minimal_surface|minimal surfaces]], 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. [[:en:Harmonic_coordinates|Harmonic coordinates]] are a harmonic [[:en:Diffeomorphism|diffeomorphism]] from a manifold to an open subset of a Euclidean space of the same dimension.
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