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. . , fk , . . .→ f ∈ D′ (Rn ),åñëè∀φ ∈ D(Rn )- ñëàáàÿ ñõîäèìîñòü∀φ(x) ∈ D(Rn ), ( fk , φ) → ( f, φ)Îïðåäåëåíèå:∫ ôóíêöèÿ f (x) íàçûâàåòñÿ ëîêàëüíî èíòåãðèðóåìîé,åñëè ∀B > 0| f (x)|dx ñóùåñòâóåò è êîíå÷åí.|x|<B∫( f, φ) =∫f (x)φ(x)dx =Rnf (x)φ(x)dx|x|<A(δ(x), φ(x)) = φ(0)Îáîáùåííûå ôóíêöèè ìîæíî äèôôåðåíöèðîâàòü áåñêîíå÷íîå ÷èñëî ðàç.f (x) ∈ Cp (Rn ) ⇒ f α (x) ∈ C(Rn ) ∀|α| 6 p∫∫f (x)Dα φ(x)dx = (−1)|α| ( f, Dα φ(x)),(F, φ) = [Dα f (x)]φ(x)dx = (−1)|α|îòñþäà:RnRn(Dα f, φ) = (−1)|α| ( f, Dα φ(x))(∆ f, φ) =(∑nk=1) ∑)nn ( 2n( ∂2 φ ) ( ∑∂ f ) ∑∂2∂22f, φ =,φ =(−1) f, 2 = f,φ = ( f, ∆φ)∂x2k∂x2k∂xk∂x2kk=1k=1k=11Âåðí¼ìñÿ ê äîêàçàòåëüñòâó òåîðåìû. K3 = − 4π|x|, âîçüì¼ì ïðîèçâîëüíóþ φ(x) ∈ D(Rn )Ω = {x : |x| < A + 1}∫ (I)1φ(0) =−∆ y φ(y)dy +4π|y||y|<A+1|y|=A+1I()∂1φ(y) →−dS y −4π|y|∂−ny(−|y|=A+1Âòîðîå è òðåòüå ñëàãàåìûå ðàâíû 0.) ∂φ(y)1· − dS y4π|y| ∂→ny) ( () )()111, ∆x φ(x) = ∆ −,φ ⇒ ∆ −= δ(x)(δ(x), φ(x)) = −4π|x|4π|x|4π|x|(K3- ôóíäàìåíòàëüíîå ðåøåíèå îïåðàòîðà Ëàïëàñà.Òåîðåìà 8.2: âñÿêàÿ u(x), ãàðìîíè÷åñêàÿ â îáëàñòè Ω, áåñêîíå÷íî äèôôåðåíöèðóåìà â Ω, òî åñòüu(x) ∈ C∞ (Ω)00Ðàññìîòðèì)ïðîèçâîëüíóþ òî÷êó x ∈ Ω.

Òîãäà ∃B(x , r) ⊂ Ω(u(x) ∈ C2 B(x0 , r))(II∂1dS y −u(x) =u(y)· →−4π|x − y|∂−ny|y−x0 |=r|y−x0 |=r()∂u(y)1−· →dS y4π|x − y| ∂−ny0<δ<rB(x0 , δ) = {x : |x − x0 | 6 δ} S(x0 , r) = {y : |y − x0 | = r}22à)|x( − y| > r − δ) > 0()á) − 4π|x1− y| ∈ C∞ B(x0 , δ) × S(x0 , r))∂u(y)1â)u1 (x) =−· →dS y4π|x − y| ∂−ny|y−x0 |=r()I∂u(y)∂1−· − dS yũ1 (x) =∂x14π|x − y| ∂→ny(I|y−x0 |=r∂u1= ũ1∂x1)[)]((I3∑∂∂11dS y =· nk (y)· u(y) · dS yu(y)· →−−4π|x − y|4π|x − y|∂yk∂−nyk=1 ñèëó òåîðåìû îá èíòåãðàëå ñ ïàðàìåòðîìIu2 (x) =ũ2 (x) =|y−x0 |=rI3∑[|y−x0 |=r(∂1∂−4π|x − y|∂x1 ∂ykk=1)]· nk (y)· u(y) · dS y ,àíàëîãè÷íî|y−x0 |=ryk − x0k; u(y)|y − x0 |∂u2= ũ2∂x1nk =∈ C(S(x0 , r))() ∂u(y)()1Dαx −∈ C B(x0 , δ) × S(x0 , r)→−4π|x − y| ∂n yI) ∂u(y)(1dS yDαx −−4π|x − y| ∂→nyÒî åñòü ìû äîêàçàëè, ÷òî â êàæäîé òîêå Ω ôóíêöèÿ áåñêîíå÷íîå ÷èñëî ðàç äèôôåðåíöèðóåìà, ÷òîè òðåáîâàëîñü.Òåîðåìà 8.3:(Î ñðåäíåì)Ïóñòü u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â B(x0 , r) è u(x) ∈ C1 (B(x0 , r))I1u(y)dS yÒîãäà u(x0 ) = 4πr2|y−x0 |=rèñïîëüçóåì èíòåãðàëüíîå ïðåäñòàâëåíèå:Iu(x ) =0|y−x0 |=r(|y−x0 |=r)()∂u(y)1· →dS y−04π|x − y| ∂−nyI∂11u(y)· →u(y)dS y−dS=y4π|x0 − y|4πr2∂−ny|y−x0 |=r|y−x0 |=r)III (∂u(y)∂u(y)111−n , ∇u)dS =(→2)−· − dS y =dS y =y→−4πr4πr4π|x0 − y| ∂→ny∂n y000|y−x |=r|y−x |=r|y−x |=r∫∫11=du(∇u)dy =∆u(y)dy = 04πr4πr1)I()I∂1u(y)· →−dS y −4π|x0 − y|∂−ny|y−x0 |6r|y−x0 |6rÇäåñü ìû èñïîëüçîâàëè ôîðìóëà Îñòðîãðàäñêîãî-Ãàóññà.Òåîðåìà: Ïóñòü u(x1 , x2 , x3 ) - íåïðåðûâíàÿ ôóíêöèÿ â Ω è1u(x ) =4πr2I0Ýòî ðàâíîñèëüíî òîìó, ÷òî u - ãàðìîíè÷åñêàÿ ôóíêöèÿ.u(y)dS y∀x0 ∈ Ω, ∀r.|y−x0 |=rÒåîðåìà 8.4: (Ïðèíöèï ìàêñèìóìà è ìèíèìóìà.) Åñëè ãàðìîíè÷åñêàÿ ôóíêöèÿ äîñòèãàåò â íåêîòîðîé òî÷êå îáëàñòè G max èëè min, òî u(x) = const ∀x ∈ Ω1)Ïóñòü u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â Ω, u(a) = maxu(x)x∈ΩÄîêàæåì òîãäà,÷òî u(x) ≡ const = u(a) ∀x ∈ B(a, d), ãäå d - ðàññòîÿíèå îò òî÷êè a äî ãðàíèöû ΩÐàññìîòðèì çàìêíóòûé øàð B(a, r);1u(a) =4πr2I|y−a|=r1u(y)dS y =4πr2I|y−a|=rr<du(a)[u(y) − u(a)]dS y +4πr223IdS y ⇒|y−a|=r14πr2I[u(y) − u(a)]dS y = 0|y−a|=rÍî ïîäûíòåãðàëüíîå âûðàæåíèå 6 0.

Ýòî âîçìîæíî ëèøü òîãäà,êîãäà u(y) = u(a)∀y : |y − a| = r øàðå ôóíêöèÿ òîæäåñòâåííî ðàâíà u(a).Äàëåå ðàññìîòðèì ∀b ∈ Ωγ∈Ω- êîíòóð, ñîåäèíÿþùèé òî÷êè a è bÏóñòü ýòîò êîíòóð èìååò ïàðàìåòðèçàöèþ x = x(s) (x1 = x1 (s), x2 = x2 (s), x3 = x3 (s))Ïóñòü d˜ = dist{}γ; R3 \Ω > 0Ðàçîáü¼ì íàø êîíòóð: 0 = s0 < s1 < s2 < . . . < sN = l;|x − x | 6 sk − sk−1{˜ ∈ΩB(xk−1 , d)˜xk ∈ B(xk−1 , d)kk−1< d˜(sk − sk−1 ) < d;˜ xk = x(sk ), k = 0, N1)Âûáåðåì x0 , B(x0 , d)˜ ⇒ u(x1 ) = u(a)2) Äàëåå ïðîäåëûâåì àíàëîãè÷íûé òðþê äëÿ âòîðîãî øàðà.

 èòîãå ìû äîáåð¼ìñÿ äî òî÷êè bÄîêàçàòåëüñòâî çàâåðåøíî.Ñëåäñòâèå èç Òåîðåìû 8.4: Ïóñòü1)Ω - îãðàíè÷åííàÿ îáëàñòü â R32)u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â Ω3)u(x) ∈ C(Ω)Òîãäà: minu(y) 6 u(x) 6 max u(y)y∈∂Ωy∈∂Ω∃a ∈ Ω, òàêàÿ ÷òî u(x) 6 u(a)Çàìå÷àíèå: Ïóñòü âûïîëíåíû óñëîâèÿ ñëåäñòâèÿ òåîðåìû 8.4. Òîãäà |u(x)| 6 max|u(y)|y∈∂ΩÄåéñòâèòåëüíî:u(x) 6 max u(y) 6 max |u(y)|y∈∂Ωy∈∂Ω−u(x) 6 max(−u(y)) 6 max |u(y)|y∈∂Ωy∈∂Ω{(∗)3∆u = f (x), x ∈ Ω ⊂ RuΓ = u0 (x), x ∈ ∂Ω = ΓΩ − îãàíè÷åííàÿîáëàñòüÊëàññè÷åñêèì ðåøåíèåì íàçûâàåòñÿ u(x) ∈ C2 (Ω) ∩ C(Ω), óäîâëåòâîðÿþùåå (∗)Òåîðåìà 8.5: Íå ìîæåò ñóùåñòâîâàòü áîëåå îäíîãî êëàññè÷åñêîãî ðåøåíèÿ çàäà÷è Äèðèõëå äëÿóðàâíåíèÿ Ïóàññîíà.Ïóñòü uI (x) è uII (x) - ðåøåíèÿ.

v(x) = uI (x) − uII (x){∆v(x)=0vΓ = 0|v(x)| 6 max |v(y)| = 0 ⇒ v(x) ≡ 0y∈∂ΩÏóñòü Ω - îãðàíè÷åííàÿ îáëàñòü â R3 . Ïóñòü u(x) ∈ C2 (Ω) ∩ C1 (Ω)Ïóñòü ∫u(x)( óäîâëåòâîðÿåò(∗) è ∆u = f (x) ∈ C(Ω))()IIu(x) =Ω1−· ∆ y u(y)· dy +4π|x − y||{z}∂Ω|∂1u(y)· →−dS y −4π|x − y|∂−ny{z}(−∂Ω|)∂u(y)1· →dS y4π|x − y| ∂−ny{z}ïîòåíöèàë äâîéíîãî ñëîÿ ïîòåíöèàë ïðîñòîãî ñëîÿîáü¼ìíûé ïîòåíöèàëÏîòåíöèàë ïðîñòîãî ñëîÿ íàì î÷åíü ìåøàåò. Õîòåëîñü áû îò íåãî èçáàâèòüñÿ.Ãðàíèöà êëàññà C2K3 (x, y) =−14π|x − y|24g(x, y) ∈ C2 (Ω) ∩ C1 (Ω)∆ g(x, y) = 0 ∀y ∈ Ω y1g(x,y)=y∈∂Ω4π|x − y|G(x, y) = −1+ g(x, y)4π|x − y|Ïóñòü ìû íàøëè g(x, y), òîãäà ìû ìîæåì ïðèìåíèòü âòîðóþ ôîðìóëó Ãðèíà:∫∫∆ y u(y)g(x, y)dy −ΩΩ∫I∂u(y)∂g(x, y)u(y) ∆ y g(x, y) dy =g(x, y) →dS−u(y)y−→− dS y∂n y|{z} ∂n y| {z }∂Ω∂Ω= u0 (y)=0Ñêëàäûâàÿ ýòó ôîðìóëó, è äðóãóþ áîëüøóþ ôîðìóëó íà ýòîé ñòðàíèöå, èìååì:∫ []][I1∂1+ g(x, y) · ∆ y u(y)· dy +u(y)· →+g(x,y)dS y −−4π|x − y|4π|x − y|∂−nyΩ∂Ω[]I∂u(y)1−· −+ g(x, y) dS y−4π|x − y|∂→nyu(x) =−∂ΩÒàê êàê ìû òàê ñïåöèàëüíî âûáèðàëè ôóíêöèþ g(x, y), òî:∫u(x) =ΩI∂u(y) →G(x, y) ∆ y u(y) dy +− G(x, y)dS y|{z} ∂n y| {z }∂Ω= u0 (y)= f (y)G(x, y) = G(y, x) ∀x, y ∈ ΩÏîïðîáóåì ðåøèòü ñëåäóþùóþ ñèñòåìó:∆ y g(x, y) = 0, y ∈ Ω 1g(x.y)=+|y|=R4π|x − y| |y|=Rx∗ = xR2|x|2- òî÷êà ñèììåòðè÷íÿ x îòíîñèòåëüíî îêðóæíîñòè.R; x,0 4π|x||x∗ − y|g(x, y) = 1; x=04πR∀y ∈ ΩÄîêàæåì, ÷òî ýòî äåéñòâèòåëüíî áóäåò ðåøåíèåì ñôîðìóëèðîâàííîé ñèñòåìû.|x||x∗ | = |y|2|y||x|= ∗|y| |x ||y − x||x|=∗|y − x | |y|⇒R1=∗4π|x||y − x | 4π|y − x|Ðåøåíèå â ñëó÷àå øàðà:{∆u(x)= 0, x ∈ Ωu= u0 (x)|x|=Ry→−ny =R- âåêòîð âíåøíåé íîðìàëè.[]33∑∑∂G(x, y) 1R∂∂−+=G(x, y) =nk (y)=nk (y)−∂yk |y|=R∂yk4π|x − y| 4π|x||x∗ − y|∂→nyk=1k=1[[33∗ ]∗ ]1 ∑ yk (yk − xk ) R (yk − xk )1 ∑ yk yk − xk |x|2 (yk − xk )·=−=−·4πR |x − y|3|x| |y − x∗ |34πR |x − y|3R2 |y − x|3k=1k=1251|x|=⇒∗|y − x | R|x − y|[[]]∂1|x|21|x|2 2 |x|22∗) =∗) =G(x,y)=(y,y−x)−(y,y−x|y|−(y,x)−|y|+(y,x−4π|x − y|3 RR24π|x − y|3 RR2R2∂→ny[2(2 )]221|x|RR − |x|=R2 − (y, x) − |x|2 + 2 y, x 2 =34π|x − y| RR|x|4πR|x − y|3Ïîñëåäíåå ðàâåíñòâî ñëåäóåò èç òîãî, ÷òî:Îòñþäà èìååì ôîðìóëó Ïóàññîíà:1u(x) =4πRI|y| = R(R2 − |x|2 )u0 (y)dS y|x − y|3(∗∗)Ïîäâåä¼ì äîêàçàííûå òîëüêî ÷òî ôàêòû â òåîðåìó:Òåîðåìà 8.6: ïóñòü u0 (x) ∈ C(Γ), ãäå Γ- ñôåðà ðàäèóñà R : Γ = {x : x ∈ R3 ;îïðåäåëÿåìîé ôîðìóëîé Ïóàññîíà:1)u(x) ∈ C∞ ({|x| < R}) ∩ C({|x| 6 R})2)u ÿâëÿåòñÿ êëàññè÷åñêèì ðåøåíèå çàäà÷è Äèðèõëå äëÿ óðàâíåíèÿ Ëàïëàñà.3)Ýòî ðåøåíèå åäèíñòåííîÒåîðåìû Ëèóâèëëÿ è îá óñòðàíèìîé îñîáåííîñòè.|x| = R},òîãäà u(x),Òåîðåìà 8.7: (Ëèóâèëëÿ) Ãàðìîíè÷åñêàÿ ôóíêöèÿ u(x) â R3 , èìåþùàÿ íà ∞ ðîñò íå âûøå ñòåïåííîãî: |u(x)| 6 C(1 + |x|)µ ∀x ∈ R3 ÿâëÿåòñÿ ìíîãî÷ëåíîì ïåðåìåííûõ x1 , x2 , x3 ñòåïåíè íå âûøå µ :(C > 0, µ - äåéñòâèòåëüíûå ÷èñëà)Ïåðâûé ñëó÷àé: µ > 0R > 0; R > 2|x| u(x) ∈ C∞ (R3 )I1R2 − |x|2u(x) = 4πRu(y)dS y|x − y|3|y|=R|y|=RRR|x| 6& |y| = R ⇒ |x − y| > |y| − |x| >22( 2)I2R − |x|1DαxDα u(x) =u(y)dS y4πR|x − y|3(DαxR − |x||x − y|322)|y|=R=Pα (R, x, y)|x − y|3+2|α|Pα (R, x, y) - îäíîðîäíûé ìíîãî÷ëåí ñòåïåíèìåòîäîì ìàòåìàòè÷åñêîé èíäóêöèè.|α| + 2ïåðåìåííûõR, x1 , x2 , x3 , y1 , y2 , y3 .Äîêàæåì ýòî1)α = 0 ⇒ |α| = 0(Dαx)R2 − x21 − x22 − x23R2 − |x|2=|x − y|3|x − y|32)Ïóñòü ïðåäïîëîæåíèå èíäóêöèè âåðíî äëÿ |α| 6 k (k > 0).

Äîêàæåì, ÷òî îíî âåðíî äëÿ |α| = k + 1Ïóñòü íå óìîëÿÿ îáùíîñòè: (α1 + 1, α2 , α3 ) = α̂(ñëó÷àè (α1 , α2 + 1, α3 ) = α̂ è (α1 , α2 , α3 + 1) = α̂ àíàëîãè÷íû)∂))[ ( 2)]((Pα (R, x, y))· |x − y|2 − (3 + 2|α|)· Pα (R, x, y)· (x1 − y1 )222P(R,x,y)R−|x|∂R−|x|∂∂x1ααα̂=Dx=D==∂x1 x |x − y|3∂x1 |x − y|3+2|α||x − y|3|x − y|3+2(|α|+1)(Pα̂ (R, x, y)|x − y|3+2|α̂|R& ∀|y| = R & ∀α = (α1 , α2 , α3 )2)∀|x| 62 (2 α R − |x|2 )CαDx 6 1+|α|3|x − y|R=à)|Pα (R, x, y)| 6 Cα R|α|+226òàê êàê ýòî ìíîãî÷ëåí ñòåïåíè íå âûøå ÷åì |α| + 2, è x1 , x2 , x3 , y1 , y2 , y3 âñå ïî ìîäóëþ ìåíüøå Rá)|x − y|3+2|α| 6( 1 )3+2|α|2· R3+2|α| ( α R2 − |x|2 )1Cα · R|α|+2= C̃α · |α|+16Dx()33+2|α||x − y| R1· R3+2|α|2Äîêàæåì, ÷òî ∀αI : |α| > µ Dαx u(x) ≡ 0 II( 22)(1 + R)µ11 C̃α1α R − |x|µ6|Dαx u(x)| =Du(y)dS·C·(1+|x|)dS6C̃·C·dS y =yyαx3 4πR4πR4πR|x − y|R|α|+1R|α|+1|y|=R=|y|=Rµ(1 + R)C̃α · C (1 + R)4πR2 = C· C̃α4πR R|α|+1R|α|µ→0|y|=Rïðè R → ∞Çàïèøåì ðÿä Òåéëîðà â ôîðìå Ëàãðàíäæà:u(x) = u(0) +Dα u(x) =m ∑∑ 1∑1 αD u(0)xα +Dα u(ξ)xα ,α!α!k=1 |α|=kα1α2∂∂∂α3;α1 ·α2 ·∂x1 ∂x2 ∂xα3 3ãäå:|α|=m+1xα = xα1 1 · xα2 2 · xα3 4 ;α! = α1 !α2 !α3 !Çíà÷èò íàøà ôóíêöèÿ è â ñàìîì äåëå ìíîãî÷ëåí ñòåïåíè íå âûøå, ÷åì µÂòîðîé ñëó÷àé: µ < 0|u(x)| 6 C(1 + |x|)µ ⇒òîãäà íàøà ôóíêöèÿ ýòî òîæäåñòâåííûé 0.Òåîðåìà 8.8 (Îá óñòðàíèìîé îñîáåííîñòè)Ïóñòü:1) u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â B (a, ρ) = {x, 0 < |x − a| < ρ}2)u(x) = o(K3 (x − a)) ïðè x → aÒîãäà:∃ lim u(x),è ôóíêöèÿ, äîïîëíåííàÿ ïðåäåëîì çíà÷åíèÿ â òî÷êå a - ãàðìîíè÷åñêàÿ ôóíêöèÿ â B(a, ρ)Äîêàçàòåëüñòâî: ïóñòü ñíà÷àëà a = 0x→a( )1⇒ |x|u(x) → 0 ïðè |x| → 0u(x) = o|x|Çàôèêñèðóåì r : 0 < r < ρI(r2 − |x|2 )1B(0, r)\{0} û(x) =u(y)dS y4πr|x − y|3û(x) ∈ C∞ (B(0, r)),|y|=rñîâïàäàåò ñ u(x) íà ãðàíèöå B(0, r)Ïóñòü V(x) = u(x) − û(x), 0 < |x| 6 rCâîéñòâà:1)V(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â B(0, r)\{0}2)|x|V(x) = |x|u(x) − |x|û(x) → 0, ïðè x → 0 (û(x) - îãðàíè÷åíà)Äîêàæåì, ÷òî |v(x)| 6 |x|ϵ∀0 < |x| < rϵWϵ± (x) =∓ V(x), 0 < |x| 6 r, ϵ > 0|x|±à)Wϵ (x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â 0 < |x| < rá)∀r > δ > 0 Wϵ± (x) ∈ C(δ 6 |x| 6 r)â)Wϵ± (x)|x|=r = ϵr[]ϵδ±ã)Wϵ (x)|x|=δ = δ 1 ± ϵ V(x) |x|=δ∀ϵ > 0 ∃δ(ϵ) > 0 ∀x : 0 < |x| 6 δ(ϵ) → |x||V(x)| <Òîãäà ∀x : 0 < |x| 6 δ(ϵ)27ϵ2Wϵ± (x) =[][]|x|V(x)|x|V(x)ϵϵϵ 1ϵ1∓>1−> · >0∓ V(x) =|x||x|ϵ|x|ϵ|x| 2Ïî ïðèíöèïó ìàêñèìóìà, Wϵ± (x) ïîëîæèòåëüíà âåçäå â 0 < |x| 6 r :ϵ∓ V(x) > 0 ∀0 < |x| 6 r|x|ϵ|V(x)| <⇒ V(x) ≡ 0 â ëþáîé êîíå÷íîé|x|Wϵ± (x) =òî÷êå , 0.lim u(x) = lim û(x) = û(0)x→0,x,0x→0,x,0Ðåãóëÿðíîñòü ïîâåäåíèÿ ãàðìîíè÷åñêèõ ôóíêöèé íà áåñêîíå÷íîñòè.2RÏóñòü x∗ = x |x|2x ∼ ∞; x∗ ∼ 0Ëåììà 8.2: Åñëè u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ â îêðåñòíîñòè ∞ â Rn , òî ôóíêöèÿ u∗ (x∗ ) =- ãàðìîíè÷åñêàÿ ôóíêöèÿ â îêðåñòíîñòè 0.(R|x∗ |)n−2|x| = ρ, |x∗ | = r |x∗ ||x| = ρr = R2Äîêàçàòåëüñòâî ëåììû áóäåì ïðîâîäèòü äëÿ ñëó÷àÿ n = 3.û(ρ, θ, φ) = u[ρ sin θ cos φ, ρ sin θ sin φ, ρ cos θ]û∗ (r, θ, φ) = u∗ [r sin θ cos φ, r sin θ sin φ, r cos θ]()R R2û, θ, φrr( 2)])22 [ ( 21R1 ˆ∂∗ (r, θ, φ) = R ∂ u R , θ, φ + R ∆′∗d∗ =ˆû∆u∆û[r·û(r,θ,φ)]+,θ,φθ,φr ∂r2r ∂r2rr2r2 θ,φ r( 2)∂ρρ2 ∂∂ R∂R2 ∂ûû(ρ, θ, φ)û(ρ, θ, φ) = − 2û(ρ, θ, φ), θ, φ ==− 2r∂r∂ρ∂rr ∂ρR ∂ρ( ρ2)[ ρ2] ρ2[]()R2 ∂∂2 R2∂∂∂∂ 2 ∂−û(ρ,θ,φ)=−−û(ρ,θ,φ)=ρû(ρ,θ,φ)⇒û,θ,φ=r∂r∂ρ∂r2R2 ∂ρr2 ∂ρR2 ∂ρR4 ∂ρ[]]33ρ5 [ 1 ∂ ( 2 ∂û(ρ, θ, φ) ) 1d∗ = ρ ∂ ρ2 ∂ û(ρ, θ, φ) + ρ ∆ˆ ′ û(ρ, θ, φ) =ˆ θ,φ û(ρ, θ, φ) =∆uρ+∆∂ρ∂ρR5 ∂ρR5 θ,φR5 ρ2 ∂ρρ25ρ c= 5 ∆u(ρ,θ, φ) = 0RÒàê êàê û - ãàðìîíè÷åñêàÿ.

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