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Îòñþäà ñëåäóåò ëèíåéíàÿ çàâèñèìîñòüýòèõ ôóíêöèé.Ñôîðìóëèðóåì áåç äîêàçàòåëüñòâà ñëåäóþùèå ôàêòû:1)λ1 , . . . , λn , . . . - ñîáñòâåííûå çíà÷åíèÿ, èõ ñ÷¼òíî, áîëåå òîãî λk → ∞. Òî åñòü îòñóòñòâóåò êîíå÷íàÿòî÷êà ñãóùåíèÿ.2)Òåîðåìà Ñòåêëîâà: Âñÿêàÿ ôóíêöèÿ u(x) ∈ D(A) ðàçëàãàåòñÿ â ðàâíîìåðíî è àáñîëþòíî ñõîäÿùèéñÿðÿä ïî ∞ñèñòåìå ñîáñòâåííûõ ôóíêöèé îïåðàòîðà Øòóðìà-Ëèóâèëëÿ:∑(u, Xk )- êîýôôèöèåíòû Ôóðüå.u(x) =ck Xk (x);ck =(Xk , Xk )k=13)Ñèñòåìà ñîáñòâåííûõ ôóíêöèé Xk (x) - ïîëíà â L2Êîëåáàíèÿ êðóãëîé ìåìáðàíû.utt = a2 ∆u = a2 (ux1 x1 + ux2 x2 ), t > 0; x = (x1 , x2 ) ∈ D = {|x| < R} u = u (x); u = u (t) x ∈ D0t t=01t=0 u = 0; t > 0u(t, x) = θ(t)V(x),∂Dïîñëå ïîäñòàíîâêè èìååì:{θ′′ (t) + λa2 θ(t) = 0−∆V(x) = λV(x), x ∈ D16Êàê ðåøàòü ïåðâóþ çàäà÷ó èç ñèñòåìû ìû çíàåì.
Îñòàíîâèìñÿ ïîäðîáíåå íà âòîðîé:−∆V(x) = λV(x), x ∈ D V ∂D = 0 V(x).0Ïåðåéä¼ì ê ïîëÿðíûì êîîðäèíàòàì:ãäå V[x1 , x2 ] = V[ρ cos φ, ρ sin φ] = V̂(ρ, φ)()1 ∂ ∂V̂1 ∂2 V̂ρ+ 2+ λV̂ = 0,ρ ∂ρ ∂ρρ ∂φ20 6 ρ < R; 0 6 φ 6 2πV̂(R, φ) = 0; 0 6 φ 6 2πÅù¼ íàäî ïîçàáîòèòüñÿ î 2π ïåðèîäè÷íîñòè:V̂(ρ, φ) . 0V̂(ρ, φ) = V̂(ρ, φ + aπ),06ρ6RÒåïåðü ó íàñ ïðÿìîóãîëüíàÿ îáëàñòü è èù¼ì ðåøåíèå ïî ìåòîäó Ôóðüå.V̂(ρ, φ) = U(ρ)Φ(φ)11U Φ + U′ Φ + 2 UΦ′′ + λUΦ = 0ρρU′′ 1 U′Φ′′Φ′′++λ=− 2⇒ −=µUρUΦρΦ′′Òî åñòü íàøà çàäà÷à îïÿòü ðàçáèâàåòñÿ íà äâå: ′′Φ (φ) + µΦ(φ) = 0Φ(φ) = Φ(φ + 2π) Φ′ (φ) = Φ′ (φ + 2π)(µ)1 ′ U′′ (ρ) + U (ρ) + λ − 2 U(ρ) = 0ρρ U(R) = 0Ñíà÷àëà ðåøàåìïåðâóþ ñèñòåìó:√√Φ(φ) = d1 cosÝòà ôóíêöèÿµφ + d2 sin µφáóäåò 2π ïåðåîäè÷åñêîé,òîëüêî åñëè µ = n2 , n ∈ ZΦ(φ) = d1 cos nφ + d2 sin nφÒîãäà âòîðàÿ ñèñòåìà çàïèñûâàåòñÿ â âèäå:)(n21 ′′′U(ρ) = 0U(ρ)+λ−U(ρ)+ρρ2U(R) = 0 |U(0)| < ∞Ïóñòü U(ρ) = W(√λρ) = W(t)Òîãäà: W′′ (t) + 1t W′ (t) + (1 − νt2 )W(t) = 02Îïðåäåëåíèå: óðàâíåíèå âèäà W′′ (t) + 1t W′ (t) + (1 − νt2 )W(t) = 0, t > 0 íàçûâàåòñÿ óðàâíåíèåìÁåññåëÿ èíäåêñà ν.
 îáùåé òåîðèè ν êîìïëåêñíîå, íî â äàííîì ñëó÷àå áóäåì ðàññìàòðèâàòü òîëüêîäåéñòâèòåëüíûå íåîòðèöàòåëüíûå ν2Åãî ðåøåíèå âîîáùå ãîâîðÿ íå âûðàæàåòñÿ ÷åðåç ýëåìåíòàðíûå ôóíêöèè.2Ïðè ìàëûõ t èìååì: W̃′′ (t) + 1t W̃′ (t) − νt2 W̃(t) = 0 - óðàâíåíèå Ýéëåðà.Ðåøàåì ýòî óðàâíåíèå:W̃(t) = α· tβ[β(β − 1) + β − ν2 ]α· tβ−2 = 0 ⇒ β2 − ν2 = 0 ⇒ β = ±ν- âåðíî, åñëè ν > 0Ðåøåíèå èùåì â âèäåîáîáùåííîãîñòåïåííîãî ðÿäà.
 0 êîíå÷åí, áåð¼ì W̃I (t):∞W̃I (t) = tν ; W̃II (t) = t−νW(t) = tν · ψν (t) = tν(∑C p tpp=0)ν2tW ′′ + W ′ + t −W=0t17()ν2(tW ′ )′ + t −W=0t(t(tW ′ )′ + t2 − ν2 )W = 0′tW (t) = ν· tt(tW ′ )′ = tνtν∞∑·t∞∑∞∑νCp t + tpp=0Cp · p· tp−1·t = tν∞∑(ν + p)Cp tpp=0(ν + p)2 Cp tp(ν + p)2 Cp tp + (t2 − ν2 )tν∞ [{∑∞∑p=0p=0p=0tνν−1äèâåðãåíòíàÿ ôîðìà.∞∑Cp tp = 0p=0∞}∑]Cp tp+2(ν + p)2 − ν2 Cp · tp += tν∞ [{∑p=0p=0∞}∑]p(p + 2ν) Cp · tp +Cp−2 tpp=0Ïðèðàâíèâàåì ê 0 ÷ëåíû ïðè êàæäîé ñòåïåíè t.p=0:p=1: p>2:Îòñþäà â ÷àñòíîñòè ñëåäóåò, ÷òî C2k+1 = 0;=0p=20· C0 = 0(1 + 2ν)C1 = 0p(p + 2ν)Cp + Cp−2 = 0Cp = −Cp−2p(p + 2ν)Çàìåòèì, ÷òî C0 ìû ìîæåì áðàòü ëþáûì.
Åñëè C0 = 0, òî ðåøåíèå áóäåò òðèâèàëüíûì. Ïîýòîìóïðåäïîëàãàåòñÿ, ÷òî C0 , 0C0C0= − 2·12(2 + 2ν)2 · 1· (1 + ν)C2C2C0C4 = C2·2 = −=− 3= + 2·24(4 + 2ν)2 · (2 + ν)2 · 1· 2(1 + ν)(2 + ν)C4C4C0C6 = C2·3 = −=− 2= − 2·36(6 + 2ν)2 · 3(3 + ν)2 · 1· 2· 3(1 + ν)(2 + ν)(3 + ν))C2(k−2)(−1)k C0C2k = − 2= 2k2 · k· (k + ν) 2 · 1· 2· . . . · k(1 + ν)(2 + ν) . . . (k + ν)C2 = C2·1 = −Âñïîìíèì ôàêòû èç ìàòåìàòè÷åñêîãî àíàëèçà:∫+∞Γ(s) =ts−1 e−t dt (s > 0)- Ãàììà ôóíêöèÿ Ýéëåðà.ż ñâîéñòâà:01)∀s > 0 :Γ(s + 1) = s· Γ(s)2)Γ(n + 1) = n! - ëåãêî äîêàçûâàåòñÿ ïî èíäóêöèè (áàçà: Γ(1) =Îáû÷íî áåðóò C0 = 2ν Γ(ν1 + 1)(−1)kC2k = ν+2k2Γ(k + 1)Γ(ν + k + 1)∞( )2k( )ν ∑(−1)ktÒîãäà Jν (t) = W(t) = 2tΓ(k + 1)Γ(ν + k + 1) 2k=0∞∞()∑(−1)kt 2k ∑ψ(t) ==uk (t)Γ(k + 1)Γ(ν + k + 1) 2k=1k=0√|t| 6 λR = M (Íàøå ìíîæåñòâî îãðàíè÷åíî)∫∞e−t dt = 1)0Âîñïîëüçóåìñÿ ïðèçíàêîì Äàëàìáåðà: ( )2|uk+1 (t)| Γ(k + 1)Γ(k + ν + 1) t 2M1 6=·= qk → 0uk (t)Γ(k + 2)Γ(k + ν + 2) 2(k + 1)(k + ν + 1) 2ïðè k → ∞1)Ðÿä ñõîäèòñÿ àáñîëþòíî â êàæäîé òî÷êå ìíîæåñòâà.2) ëþáîé îãðàíè÷åííîé îáëàñòè, ïðèíàäëåæàùåé ìíîæåñòâó ðÿä ñõîäèòñÿ ðàâíîìåðíî.18Çàìå÷àíèå: ν = 0 - ìîæåò âûïàäàòü, íî âñ¼ õîðîøî.Ïîëó÷åííûå ôóíêöèè Jν (t) íàçûâàþò ôóíêöèÿìè Áåññåëÿ 1-îãî ðîäà (ν > 0){Yν (t) =const1 · t−ν [1 + o(1)], ν > 0const2 · ln t[1 + o(1)], ν = 0Ýòî òàê íàçûâàåìûå ôóíêöèè Áåññåëÿ âòîðîãî ðîäà èíäåêñà νÏðèáëèçèì óðàâíåíèå Áåññåëÿ â áåñêîíå÷íî óäàë¼ííîé òî÷êå.
Ââåä¼ì òàêóþ çàìåíó ïåðåìåííîãî,÷òîáû óðàâíåíèå íå ñîäåðæàëî ïåðâîé ïðîèçâîäíîé:1W(t) = √ Z(t)t√1tW ′ = tZ′ − √ Z2 t11 ′′111′ ′(tW ) = √ Z + √ Z′ − √ Z′ +√ Ztt2t t2t t4t2 t[]ν2 − 1/4′′Z (t) + 1 −Z(t) = 0t2Z(t)- ëèíåéíàÿ êîìáèíàöèÿ sin è cos, åñëè ν = 12Z̃′′ (t) + Z̃(t) = 0 ⇒ Z̃(t) = A cos(t + α)( )1Z(t) = A cos(t + α) + θ- ìîæíî ïîêàçàòü ÷òî ýòî(t)A cos(t + α)1+ θ 3/2ïðè t → ∞W(t) =√tt√)()(21π πJν (t) =+ θ 3/2ïðè t → ∞cos t − ν −πt24tòàê èñïîëüçóþ òåîðèþ âîçìóùåíèé.Íóëè êîñèíóñà ïðîñòûå, êðàòíîñòè 1 (ïðîèçâîäíàÿ â íèõ â 0 íå îáðàùàåòñÿ), èõ áåñêîíå÷íî ìíîãî.(ν)3ππ + ν + πk, k = 1, 2, . . .42áîëüøèõ t îíè ñòàíîâÿòñÿ íàµk vÏðèïðèìåðíî îäèíàêîâûõ ðàññòîÿíèÿõ.
Äëÿ ôóíêöèé Áåññåëÿ ñ ðàçëè÷íûìè èíäåêñàìè íóëè ïåðåìåæàþòñÿ.Òåïåðü çàôèêñèðóåì ν.Jν (t); µ > 0 → eJν (t) = Jν (µt)[] ()ν2d dJν (µt)t+ µ2 t −Jν (µt) = 0dtdttµ1 > 0, µ2 > 0([tJν′ (µ1 t)]′ + µ21 t −([tJν′ (µ2 t)]′ + µ22 t −→ Jν (µ1 t), Jν (µ2 t))ν2Jν (µ1 t) = 0t2)ν2Jν (µ2 t) = 0t2Äîìíîæèì ïåðâîå óðàâíåíèå íà −Jν (µ2 t), âòîðîå íà Jν (µ1 t) è ñëîæèì èõ.{[}dJν (µ2 t)dJν (µ1 t) ]dt Jν (µ1 t)·− Jν (µ2 t)·= (µ21 − µ22 )t· Jν (µ1 t)· Jν (µ2 t)dtdtdt[]1µ1 · Jν (µ2 )· Jν′ (µ1 ) − µ2 · Jν (µ1 )· Jν′ (µ2 ) ,∫122 (µ2 − µ1 ))t· Jν (µ1 t)· Jν (µ2 t)dt = ]2 1 (]2ν2 [1[ ′J(µ)+1−Jν (µ1 ) , µ1 = µ21 2 ν022µ1V̂(ρ, φ) = U(ρ)· Φ(φ)√Un (ρ) = Jn ( λρ)√Un (R) = 0 ⇒ Jn ( λR) = 0 ⇒(λnk =√(n)λR = µk(n) )2µkRV̂0k (ρ, φ) = J0(µ0kR)ρ , k = 1, 2, . .
.(λ0k =(0) )2µkR19µ1 , µ2(1V̂nk(ρ, φ)= JnµnkR)(ρ cos nφ2V̂nk(ρ, φ)= JnµnkR)(ρ sin nφλnk =(n) )2µkk, n ∈ N;RÒî åñòü â äàííîì ñëó÷àå ñîáñòâåííûå ÷èñëà áóäóò äâóêðàòíûìè.−∆V = λV V =0|x|=R V.0(∗)Ñèñòåìà ôóíêöèé îðòîãàíàëüíà îòíîñèòåëüíî ñêàëÿðíîãî ïðîèçâåäåíèÿ ïðîñòðàíñòâà L2∫2π"(u, v) =(∫RJ0u(x)v(x)dx =|x|<R∫Rû(ρ, φ)v̂(ρ, φ)ρ· dρdφ0) ( (0) )µmρ · J0ρ ρ· dρ = 0RR0(0)µkÑîáñòâåííûå ôóíêöèè, ñîîòâåòñòâóþùèå ðàçëè÷íûì ñîáñòâåííûì çíà÷åíèÿì îðòàãîíàëüíû åñëèìåíÿåòñÿ òîëüêî k.0Åñëè ìåíÿåòñÿ n, òî òàì cos nφ cos mφ, è èíòåãðàë îò ýòîé ôóíêöèè = 0⇒ íàø íàáîð ôóíêöèé îðòàãîíàëåí â ïðîñòðàíñòâå L2Ïðèâåä¼ì äðóãîé ñïîñîá äîêàçàòåëüñòâà,äëÿ ýòîãî äîêàæåì ñèììåòðè÷åíñòü îïåðàòîðà:A : D(A) = {u(x, y) ∈ C2 (Q) ∩ C1 (Q); uΓ = 0}A : u(x, y) → −∆u(x, y)(∆u, v) = ...
= (u, −∆v)Òåîðåìà: Ñèñòåìà ñîáñòâåííûõ ôóíêöèé çàäà÷è (∗)- îðòàãîíàëüíûé áàçèñ â L2 (D), òî åñòü ∀û(ρ, φ) ∈ðÿä Ôóðüå ñõîäèòñÿ ê û(ρ, φ) â íîðìå ïðîñòðàíñòâà L2 (D)Òåîðèÿ ñôåðè÷åñêèõ ôóíêöèé.L2 (D)Óðàâíåíèå Ëàïëàñà-Ïóàññîíà â R3∆u(x) =∂2 u ∂2 u ∂2 u++= f (x), x ∈ Ω ⊂ R3∂x21 ∂x22 ∂x23Îïðåäåëåíèå: Ôóíêöèÿ u(x) íàçûâàåòñÿ ãàðìîíè÷åêîé â Ω ⊂ R3 , åñëè:1)u(x) ∈ C2 (Ω)2)∆u(x) ≡ 0 ∀x ∈ Ωu(x1 , x2 , x3 ) → û(ρ, θ, φ)()[]1∂û1∂2 û21 ∂sin θ+·∆û(ρ, θ, φ) = ûρρ + ûρ + 2ρ∂θρ sin θ ∂θsin2 θ ∂φ2|{z}îïåðàòîð Ëàïëàñà-Áåëüòðàìèðåøåíèåì áóäåò: C1 + Cρ2Ëåììà 8.1:(Èíòåãðàëüíîå ïðåäñòàâëåíèå ðåøåíèÿ óðàâíåíèÿ Ïóàññîíà.)Ω - îãðàíè÷åííàÿ îáëàñòü â R3 ñ ãëàäêîé (êóñî÷íî-ãëàäêîé) ãðàíèöåé Γ:Ïóñòü u(x) ∈ C2 (Ω) ∩ C1 (Ω)∆u(x) = f (x) ∈ C(Ω).
Òîãäà ∀x ∈ Ω())∫ (II (2ûρρ + ûρ = 0ρu(x) =−Ω|1· f (y)· dy +4π|x − y|{z}1∂−u(y)· →dS y −4π|x − y|∂−nyΓΓ|{z} |−)∂u(y)1dS y· →4π|x − y| ∂−ny{z}ïîòåíöèàë äâîéíîãî ñëîÿ ïîòåíöèàë ïðîñòîãî ñëîÿîáü¼ìíûé ïîòåíöèàëÇàôèêñèðóåì x ∈ Ω ∃ϵ0 : B(x, ϵ0 ) ∈ Ω; ∀ϵ > 0; ϵ < ϵ0 B(x, ϵ) ⊂ ΩÏóñòü Ωϵx = Ω\B(x, ϵ)δϵx - ñôåðà ðàäèóñà ϵ ñ öåíòðîì â x.∂Ωϵx = Γ ∪ δϵx20Ôóíêöèÿ K3 (x, y) = − 4π|x1− y| áåñêîíå÷íî äèôôåðåíöèðóåìà è ãàðìîíè÷åñêàÿ â Ωϵx .Çíà÷èò ìîæíî âîñïîëüçîâàòüñÿ âòîðîé ôîðìóëîé Ãðèíà.()()()∫I∂u(y)111dy −dy =−dS y −∆u(y)· −u(y)· ∆ y −−4π|x − y|4π|x − y|4π|x − y|∂→nyϵϵΓΩΩx()()()I xII∂u(y)111∂∂−−−−dS−u(y)·dS+·dS yu(y) →yy−−4π|x − y|4π|x − y|4π|x − y|∂−ny∂→ny∂→nyΓδϵxδϵx()1∆y −= 0 - â ýòîì ìîæíî óáåäèòüñÿ ïðîñòîé ïðîâåðêîé.4π|x − y|)())∫ (II (∂u(y)111∂−∆u(y)· dy +−dS y −−dS y =u(y) →−−4π|x − y|4π|x − y|4π|x − y| ∂→∂n ynyϵΓΓΩ))(I (I x∂u(y)11∂dS y −+· →dS y=+u(y)· →−4π|x−y|4π|x−y|∂n y∂−nyϵϵδxδx|{z} |{z}I2 (ϵ)I1 (ϵ) ∂u(y) →−→− → = (∇u, n ) 6 |∇u|· | n | 6 M−∂n y I ()∫∂u(y)4πϵ211 1·dS=M→ 0 ïðè ϵ → 0· →|I1 (ϵ)| = dS6Myy4π|x − y| ∂−4π ϵ4πϵnyϵ|x−y|=ϵδx)(( )1∂∂ 1 1= 2=−→−|x−y|∂ρρϵ∂n yρ=ϵIIIu(x)11[u(y) − u(x)]dS y = u(x) + eI2 (ϵ)|I2 (ϵ) =u(y)dS y =· dS y +4πϵ24πϵ24πϵ2ϵϵϵδxδxδx II[] 11e|I2 (ϵ)| =u(y) − u(x) dS y 6· max |u(y) − u(x)|· dS y 6 max |u(y) − u(x)| → 0 4πϵ2 y:|x−y|=ϵ|y−x|6ϵ4πϵ2 ϵδxδϵx∫ïðè ϵ → 0Ïîäâîäÿ èòîãè, èìååì äîêàçàòåëüñòâî ëåììû.Ñëåäñòâèå èç Ëåììû 8.1: Ïóñòü Ω - îãðàíè÷åííàÿ îáëàñòü â R3 c ãëàäêîé (êóñî÷íî-ãëàäêîé) ãðàíèöåé Γ u(x) ∈ C2 (Ω) ∩ C1 (Ω); u(x) - ãàðìîíè÷åñêàÿ ôóíêöèÿ.
Òîãäà:Iu(x) =Γ())I (∂u(y)∂11u(y)· →−dS−−· − dS yy4π|x − y|4π|x − y| ∂→∂−nynyΓ1Òåîðåìà 8.1 Ôóíêöèÿ K3 (x) = − 4π|x|ÿâëÿåòñÿ ðåøåíèåì â îáîáùåííûõ ôóíêöèÿõ ñëåäóþùåãî óðàâíåíèÿ: ∆K3 (x) = δ(x), x ∈ R3∫åñëè u(x) ∈ D(A), òî u(x)uk (x)dx îäíîçíà÷íî îïðåäåëÿåò u(x)∫åñëè u(x) < D(A), òî u(x)uk (x)dx îïðåäåëÿåò ôóíêöèþ ũ, îòëè÷íóþ îò u íà ìíîæåñòâå ìåðû 01.
D(Rn ) - ëèíåéíîå ïðîñòðàíñòâî ïðîáíûõ (îñíîâíûõ) ôóíêöèé.φ(x) ∈ D(Rn ), åñëè1)φ(x) ∈ C∞ (Rn )2)∃A > 0 : ∀x : |x| > Aφ(x) ≡ 0(Ôèíèòíîñòü)φ1 (x), φ2 (x), . . . , φk (x), . . . → φ(x) ∈ D(Rn ), åñëè|{z}∈ D(Rn )à)∃A > 0, òàêîå ÷òî: φk (x) ≡ 0 ∀x : |x| > A & ∀k ∈ Ná)∀α = (α1 , . . . , αn ) Dα φk (x) ⇒ Dα φ(x) (ñõîäèìîñòü ðàâíîìåðíàÿ)-21Ïðèìåðîì ïðîáíûõ ôóíêöèé ìîæåò ñëóæèòü òàê íàçûâàåìûå øàïî÷êè:ϵ2 Cϵ · e− ϵ2 −|x|2 ,ωϵ (x) = 0, |x| > ϵ|x| 6 ϵÎïðåäåëåíèå: Îáîáùåííîé ôóíêöèåé f íàçûâàåòñÿ âñÿêèé ëèíåéíûé íåïðåðûâíûé ôóíêöèîíàë íàÎáîçíà÷àåòñÿ ( f, φ)D(Rn ).à)Ëèíåéíîñòü: ( f, λφ + µψ) = λ( f, φ) + µ( f, ψ)á)Íåïðåðûâíîñòü,åñëè φk (x) → φ(x) â D(Rn ) òî ( f, φk ) → ( f, φ)Ìíîæåñòâî îáîáù¼ííûõ ôóíêöèé îáîçíà÷àåòñÿ D′ (Rn )λ f + µg = F(F, φ) = (λ f + µg, φ) = λ( f, φ) + µ(g, φ)f1 , f2 , .
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