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Íå ôàêò, ÷òî íåòó äðóãèõ ðåøåíèé, êîòîðûå íå îïèñûâàþòñÿ èíòåãðàëîì Ïóàññîíà. Äëÿ åäèíñòâåííîñòè ðåøåíèÿ, íåîáõîäèìî ââåñòèíåêîòîðûé êëàññ ôóíêöèé ñ îãðàíè÷åííûì ðîñòîì íà áåñêîíå÷íîñòè.T > 0, σ > 0. Ââåä¼ì ñëîéôóíêöèé. u(t, x) ∈ Mσ (T), åñëè:Îïðåäåëåíèå: ÏóñòüMσ (T)êëàññ1)u(t, x)2)∃A1,2∈ Ct,x(Πt ) ∩ C(ΠT )Êëàññ Òèõîíîâà:σâRn : ΠT = {(t, x) : 0 < t < T; x ∈ Rn}∀(t, x) ∈ ΠTσ=2Óòâåðæäåíèå 1: Êëàññ ôóíêöèéMσ0 (T) ⊂ Mσ1 (T)ÏóñòüT(íåïðåðûâíî ïðîäîëæàåìà âïëîòü äî ãðàíèöû.)|u(t, x)| 6 A· eα|x| ,> 0; α > 0 :òîëùèíûMσ- ëèíåéíîé ïðîñòðàíñòâî, ïðè÷¼ì åñëèσ0 > 0; σ0 6 σ1 ,òîu1 (t, x), u2 (t, x) ∈ Mσ (T)σ|u1 (t, x)| 6 A1 eα1 |x||u2 (t, x)| 6 A2 eα2 |x||µ1 u1 (t, x) + µ2 u2 (t, x)|66 (|µ1 |A1 + |µ2 |A2 )emax(α1 ,α2 )|x|u(t, x) ∈ Mσ0 (T);σ|µ1 ||u1 (t, x)| + |µ2 ||u2 (t, x)|σσ0 < σ 1 ⇒=σσ|µ1 |A1 eα1 |x| + |µ2 |A2 eα2 |x|66σAeα|x|σ0|u(t, x)| 6 Aeα|x|6 Aeα|x|σ1⇒ u(t, x) ∈ Mσ1 (T)|x|+ 214a (T−t) , t < T, x ∈ Rne(T − t)n/22Óòâåðæäåíèå 2:∀T > 0ôóíêöèÿUT (t, x) =óäîâëåòâîðÿåò îäíîðîäíîìó óðàâíåíèþ òåïëîïðîâîäíîñòè.∞nà)UT (t, x) ∈ C ({t < T, x ∈ R })[]|x|2+ 2∂UT1n1|x|24a(T−t)á)=+e∂t(T − t)n/2+1 2 (T − t)n/2 4a2 (T − t)2[]|x|2+ 2112n[4|x|24a (T−t)á)∆x UT =+e(T − t)n/2 4a2 (T − t) (T − t)n/2 [4a2 (T − t)2 ]2∂UT− a2 ∆x UT = 0∂tv(t, x) òàêîâà, ÷òî:∈ M2 (T)2n2)LU ≡ vt − a ∆x v = 0, 0 < t < T, x ∈ R ,vt=0 = 0, x ∈ RnÒîãäà ∀ϵ > 0 ∃T1 : 0 < T1 6 T :|x|2+ 214a(2T1 −t) ,|v(t, x)| 6 ϵU2T1 (t, x) = ϵ(t, x) ∈ ΠT1e(2T1 − t)n/22α|x|1)v(t, x) 6 Ae, ∀(t, x) ∈ ΠTËåììà 4.4 Ïóñòü1)v(t, x)2)ÂûáåðåìT1 = min(T,3)Âûáåðåì∀ϵ > 01);16a2 αòî åñòü ìû âûáðàëèT1íå çàâèñÿùèì îòϵ|x|+ 2ϵ224a (2T1 −t) − Aeα|x| >ω±ϵ (t, x) = ϵU2T1 (t, x) ± v(t, x) > ϵU2T1 (t, x) − |v(t, x)| > ϵU2T1 (t, x) − Aeα|x| =en/2() ](2T1 − t)[|x|2|x|2n/21 −α |x|2+ 2(2T)+−ϵϵ21>e 4a (2T1 ) − Aeα|x| =e 8a2 T1 1 −A· e 8a2 T1ϵ(2T1 )n/2(2T1 )n/2() ][1 −α |x|2(2T1 )n/2−12∃R(ϵ) :A· e 8a T16(òàê êàê ïîêàçàòåëü ýêñïîíåíòû < 0)ϵ2ω±ϵ (t, x) - äîñòàòî÷íî ãëàäêàÿ231Îáå ýòè ôóíêöèè è îïåðàòîð òåïëîïðîâîäíîñòè îáðàùàþòñÿ â 0 â ïîëîñå øèðèíîéω±ϵ (t, x) > 0 ∀(t, x) ∈ ΠT1T1 .Èç ñëåäñòâèÿ 4.4ϵU2T1 (t, x) ± v(t, x) > 0 ∀(t, x) ∈ ΠT1|v(t, x)| < ϵU2T1 (t, x), ∀(t, x) ∈ ΠT1(t∗ , x∗ )Áåð¼ì ëþáóþ òî÷êó{ut − a2 ∆x u = f (t, x), 0 < t 6 T; x ∈ Rnut=0 = u0 (x), x ∈ RnÇàäà÷à Êîøèâ ïîëîñåΠT|v(t, x)| < ϵC ⇒ v ≡ 0èç ðàññìàòðèâàåìîé ïîëîñû:Òåîðåìà 4.5:íå ìîæåò èìåòü áîëåå îäíîãî ðåøåíèÿ â êëàññå ÒèõîíîâàÏðåäïîëîæèì ïðîòèâíîå.
Ïóñòü åñòü 2 ðåøåíèÿ:v(t, x) = uI (t, x) − VII (t, x) ∈ M2 (T){uI (t, x)Lv ≡ vt − ∆x v = 0,vt=0 = 0, x ∈ RnèM2 (T)uII (t, x)(t, x) ∈ ΠTT1 .  çàìûêàíèè ïîëîñû ΠT1 v(t, x) ≡ 0 Äàëåå, òàêv(t, x) = 0, èñïîëüçóåì îïÿòü íàøó ëåììó, äëÿ ñäâèíóòîé ââåðõ çàäà÷èòàêîå N , ÷òî NT1 > 0, òî çà N òàêèõ øàãîâ ìû ïîëó÷àåì, ÷òî â èñêîìîì ñèëó òîëüêî ÷òî äîêàçàííîé ëåììû íàõîäèìêàê íà âåðõíåé ãðàíèöå ïîëîñûÊîøè.
Òàê êàê ñóùåñòâóòåñëîåv(t, x) ≡ 0Îòìåòèì òîò ôàêò,÷òî â êëàññåM3 (T)ìîæíî ïîñòðîèòü åù¼ îäíî ðåøåíèå.Îáðàòíàÿ çàäà÷à òåïëîïðîâîäíîñòè:nut − a2 uxx = 0,ut=0 = u0 (x)t < 0, x ∈ RÏðèâåä¼ì ïðèìåð, êîãäà îíà áóäåò íåêîððåêòíîé:−nÏóñòü u0 (x) = ecos nx, n ∈ N, n → ∞ Òîãäà:−a2 n2 t−nu(t, x) = ecos nx  ëþáîé òî÷êå (t, x) ñ t < 0 ïðèn → ∞ u(t, x) → ∞Ýëëèïòè÷åñêèå çàäà÷èÔîðìóëà Ãðèíà äëÿ óðàâíåíèÿ ËàïëàñàΩ - îãðàíè÷åííàÿ îáëàñòü â RnÏóñòüÎïðåäåëåíèå: Îãðàíè÷åííàÿ îáëàñòüåñëèΩíàçûâàåòñÿ îáëàñòüþ ñ ãðàíèöåéΓêëàññàC1 (ãëàäêîé),∀x0 ∈ Γ ∃:à)Äåêàðòîâà ñèñòåìà êîîðäèíàò ξ = (ξ1 , .
. . , ξn )0′n′á)Îêðåñòíîñòü U(x ) = {x : |ξ | < r, |ξ | < h} ξ = (ξ1 , . . . , ξn−1 ) òàêàÿ, ÷òî â íåé ÷àñòü ãðàíèöû′′âèìà â âèäå: ξn = F(ξ ) = F(ξ1 , . . . , ξn−1 ), |ξ | < r:1)F(0)2)F(ξ′Γïðåäñòà-=0) ∈ C1 (|ξ′ | < r)∂f∂f(0) = . . . =(0) = 0∂ξ1∂ξn−100′â)Ìíîæåñòâî U− (x ) = U(x ) ∩ {x : ξn < F(ξ )}∈Ω00′ã)Ìíîæåñòâî U+ (x ) = U(x ) ∩ {x : ξn > F(ξ )}<Ωä) ×èñëà r > 0 è h > 0 ìîæíî âûáðàòü íåçàâèñèìî îò3)òî÷êèx0 ∈ Γôîðìóëà Îñòðîãðàäñêîãî-Ãàóññà:∫Ω→−div F (x)dx =∫ (ΩËåììà 6.1: Ïóñòü)I (→−∂F∂F−n (x))dS+ ...
+dx =F (x), →x∂x1∂xnΓΩ- îãðàíè÷åííàÿ îáëàñòü â1)Ôîðìóëà Ãðèíà(1-àÿ):∫∀u(x) ∈ C2 (Ω), v(x) ∈ C1 (Ω) :I(∆u)v· dx =ΩΓRnñ ãëàäêîé ãðàíèöåé∂u−n vdSx −∂→32∫(∇u, ∇v)· dxΩΓ ∈ C1 .Òîãäà ñïðàâåäëèâî:1)Ôîðìóëà Ãðèíà(2-àÿ):∫∀u(x) ∈ C2 (Ω), v(x) ∈ C2 (Ω) :∫I(∆u)v· dx −Ω(∆v)u· dx =ΓΩ∂u−n vdSx −∂→IΓ∂v−n udSx∂→Äîêàçàòåëüñòâî:à)→−v(x), f (x)- äâà ïîëÿ↓→−div( f v)á)→−→− ↓→−→−→−→−→−= (∇, f v) = (∇, f v) + (∇, f v) = v(∇, f ) + (∇v) f = v div f + ( f , ∇u)n∑∂∆u =(ux ) = div(∇u)∂xk kk=1→−11f =( ∇u ∈) C (Ω), v ∈ C (Ω)div (∇u)v = v∆u + (∇u, ∇v), èíòåãðèðóåì ïî Ω∫∫∫I∫I∫()∂u→−(∆u· v)dx =div (∇u)v dx− (∇u, ∇v)dx = (∇u, n )vdSx − (∇u, ∇v)dx =−n vdSx − (∇u, ∇v)dx,∂→ΩΩΓΩΓΩΩòàê êàê:n∑∂u∂u→−n ) ==(∇u,nk→−∂xk∂nk=1∫I∫∂v2)(∆v)u· dx =udS−(∇v, ∇u)· dxx−n∂→∫Ω∫ΩIΓ∂u(∆u)v· dx =−n vdSx − (∇u, ∇v)· dx∂→ΓΩΩÂû÷èòàÿ èç îäíîãî âûðàæåíèÿ äðóãîå, èìååì èñêîìîå ðàâåíñòâî.Çàìå÷àíèå 1: â ïåðâîé ôîðìóëå Ãðèíà ìîæíî âçÿòüu(x) ∈ C2 (Ω) ∩ C1 (Ω), ∆u ∈ C(Ω), v(x) ∈ C1 (Ω)Ìîæíî àêêóðàòíî äîêàçàòü ýòîò ôàêò, íî çäåñü ìû îïóñòèì äîêàçàòåëüñòâî.Âíóòðåííÿÿ çàäà÷à Äèðèõëå äëÿ óðàâíåíèÿ Ïóàññîíà.ÏóñòüΩ- îãðàíè÷åííàÿ îáëàñòü ñ ãðàíèöåéôóíêöèè.
Çàäà÷à Äèðèõëå ñîñòîèò â òîì,÷òîáû{Γ ∈ C1 . Ôóíêöèÿ u0 (x) ∈ C(Γ), f (x) ∈ C(Ω)íàéòè u(x):∆u(x)= f (x), x ∈ ΩuΓ = u0 (x), x ∈ Γ- çàäàííûå(∗)Îïðåäåëåíèå Êëàññè÷åñêèì ðåøåíèåì çàäà÷è Äèðèõëå (!ÏÎÊÀ!) íàçûâàåòñÿu(x) ∈ C1 (Ω) ∩ C2 (Ω),óäîâëåòâîðÿþùàÿ óðàâíåíèþ è ãðàíè÷íûì óñëîâèÿì.Ëåììà 6.2: íå ìîæåò ñóùåñòâîâàòü áîëåå îäíîãî êëàññè÷åñêîãî ðåøåíèÿ çàäà÷è Êîøè (*)Äåéñòâèòåëüíî, ïóñòüuI (x)èuII (x)- ðåøåíèÿ;∆v(x) ≡ 0, x ∈ Ω, ïðîäîëæàåì âïëîòü äî∆v ∈ C(Ω); vΓ = 0Èñïîëüçóåìïåðâóþôîðìóëó∫I∫ Ãðèíà:∂v(∆v)v· dx =−n vdSx − (∇v, ∇v)· dx∂→ΓΩËåâàÿ ÷àñòü∫v(x) = uI (x) − uII (x) ∈ C1 (Ω) ∩ C2 (Ω)ãðàíèöû:Ω= 0, òàê êàê ∆v = 0; Ïåðâîå ñëàãàåìîå â ïðàâîé ÷àñòè ðàâíî 0, òàê êàê íà ãðàíèöå v = 0(∇v, ∇v)· dx = 0⇒|∇v(x)| = 0, ∀x ∈ Ω⇒v(x) = const, ∀x ∈ Ω,è â ñèëó òîãî,÷òî íà ãðàíèöåΩv(x) = 0,èìååì, ÷òîv(x) ≡ 0, ∀x ∈ Ω ⇒äâà ýòèõ ðåøåíèÿ íà ñàìîì äåëå ñîâïàäàþò.Âíóòðåííÿÿ çàäà÷à Íåéìàíà äëÿ óðàâíåíèÿ Ïóàññîíà.Ω- îãðàíè÷åííàÿ îáëàñòü ñ ãëàäêîé ãðàíèöåéf (x) ∈ C(Ω), u1 (x) ∈ C(Γ)Γ:Íàéòè u(x):33∆u(x) = f (x), x ∈ Ω ∂u = u0 (x), x ∈ Γ →∂−n (∗∗)ΓÎïðåäåäåíèå: Êëàññè÷åñêèì ðåùåíèåì çàäà÷è Íåéìàíà (!ÏÎÊÀ!) íàçûâàåòñÿ òàêàÿ ôóíêöèÿu(x) ∈ C1 (Ω) ∩ C2 (Ω), óäîâëåòâîðÿþùàÿ óðàâíåíèþ è ãðàíè÷íûì óñëîâèÿì.Ëåììà 6.3: Ëþáûå äâà êëàññè÷åñêèõ ðåøåíèÿ çàäà÷è Íåéìàíà îòëè÷àþòñÿ íà êîíñòàíòó.uI (x) è uII (x) - ðåøåíèÿ; v(x) = uI (x) − uII (x) ∈ C1 (Ω) ∩ C2 (Ω)∆v(x) ∈ C(Ω) ∆v ≡ 0, x ∈ Ω∂v −n = 0∂→Γ∫I∫∂v(∆v)v· dx =−n vdSx − (∇v, ∇v)· dx∂→Äåéñòâèòåëüíî, ïóñòüΓΩËåâàÿ ÷àñòüΩ= 0,òàê êàê∆v = 0;Ïåðâîå ñëàãàåìîå â ïðàâîé ÷àñòè ðàâíîv(x) = const,Àíàëîãè÷íî0,òàê êàê íà ãðàíèöå∂v−n = 0∂→÷òî è òðåáîâàëîñü äîêàçàòü.
Âåðíî è îáðàòíîå:Ëåììà 6.4: Íåîáõîäèìûì óñëîâèå âíóòðåííåé çàäà÷è Íåéìàíà (**) ÿâëÿåòñÿ âûïîëíåíèå ðàâåíñòâà:∫If (x)dx =u1 (x)dSxΓΩ∫∫f (x)dx =Äåéñòâèòåëüíî,∫ΩIf (x)dx =∫∆u(x)dx =ΩI∆u(x)· 1· dx =ΓΩ∂u−n · 1· dSx −∂→∫(∇u, ∇1)dx ⇒Ωu1 (x)dSxΓΩÎ çàäà÷å íà ñîáñòâåííûå ôóíêöèè è ñîáñòâåííûå çíà÷åíèÿ äëÿ îïåðàòîðà Ëàïëàñàïðè îäíîðîäíîì óñëîâèè Äèðèõëå.Íàéòèà)∃λèu(x) ∈ C1 (Ω) ∩ C2 (Ω), ÷òî:{−∆u(x)= λu(x)uΓ = 0; u(x) . 0ñ÷¼òíîå ÷èñëî ñîáñòâåííûõ çíà÷åíèéá)|λk |→∞ïðèk→∞(λîïåðàòîð ìèíóñ äåëüòàλ : λ1 , . . . λ k , . . .ìîãóò áûòü êîìïëåêñíûìè)â)Êàæäîìó ñîáñòâåííîìó çíà÷åíèþλkîòâå÷àåò ëèøü êîíå÷íîå ÷èñëî ëèíåéíî-íåçàâèñèìûõ ñîáñòâåí-íûõ ôóíêöèé.Ëåììà 6.5: Âñÿêîå ñîáñòâåííîå çíà÷åíèå∫I(∆u)u· dx =ΩÍà ∫ãðàíèöå−λΓ∂u−n udSx −∂→Γu = ∫u=0 ⇒(uu)· dx = −∫λk > 0(∇u, ∇u)· dxΩïåðâîå ñëàãàåìîå â ïðàâîé ÷àñòè ðàâíî(∇u, ∇u)· dx∫Ω∫ Ω2λ (|u| dx = (|∇u|2 )dx ⇒ΩΩ∫2(|∇u| )dxΩλ= ∫λâåùåñòâåííî,λ > 0,è áîëåå òîãî(|u|2 )dxΩ( çàäà÷å ÍåéìàíàÂûðàæåíèå äëÿλλìîæåò áûòü ðàâíî0)íàçûâàåòñÿ ñîîòíîøåíèåì Ðýëåÿ.34λ>00çíà÷èò:Ëåììà 6.6: Îïåðàòîð −∆ , îïðåäåë¼ííûé íà ëèíåéíîì ìíîæåñòâåD0 (−∆) = {u(x) : u(x) ∈ C1 (Ω) ∩ C2 (Ω); ∆u(x) ∈ C(Ω); uΓ = 0}ôóíêöèé:ÿâëÿåòñÿ ñèììåòðè÷íûì îòíîñèòåëüíî ñêàëÿðíîãî ïðîèçâåäåíèÿ ïðîñòðàíñòâîL2 (Ω)Äîêàçàòåëüñòâî.
Âîñïîëüçóåìñÿ âòîðîé ôîðìóëîé Ãðèíà.u(x), v(x) ∈ D0 (−∆)I(−u; v) − (u, −∆v) =Γ∂u →−−n v ds +∂→Ëåììà 6.7 Ñîáñòâåííûå ôóíêöèèλk , λm ,IΓII →−v∂u∂−v∂→uds=·0·ds+−n−n−n · 0· ds ⇒ (−u; v) = (u, −∆v).∂→∂→∂→ΓΓuk (x), um (x), ñîîòâåòñòâóþùèå ðàçëè÷íûì ñîáñòâåííûì çíà÷åíèÿìL2 (Ω).îðòîãîíàëüíû îòíîñèòåëüíî ñêàëÿðíîãî ïðîèçâåäåíèÿÄîêàçàòåëüñòâî(−∆uk , um ) = (λk uk , um ) = λk (uk , um );(uk , −∆um ) = (uk , λm um ) = λm (uk , um ) = λm (uk , um ).Òàê êàêλk , λmòî(uk , um ) = 0.Çàäà÷à Äèðèõëå äëÿ óðàâíåíèÿ Ëàïëàñà â êðóãå.Ðàññìîòðèì êðóãD = {x |x| < R}, è åãî ãðàíèöó Γ = ∂D = {x |x| = R}.{∆u(x) = f (x), x ∈ Dçàäà÷à ÏóàññîíàuΓ = u0 (x), x ∈ ΓÂâåäåì çàìåíó:∆u{â êðóãåx1 = ρ cos φ; u(x1 , x2 ) → û(ρ, φ) = u(ρ cos φ, ρ sin φ)x2 = ρ sin φ(1)ïðåîáðàçóåòñÿ ñëåäóþùèì îáðàçîì:()2c = 1 ∂ r ∂û + 1 ∂ û = ûρρ + 1 ûρ + 1 ûφφ∆u2ρ ∂r ∂rρρ ∂φ2ρ2(2)Çàïèøåì òåïåðü çàäà÷ó Äèðèõëå:11ûρρ + ûρ + 2 ûφφ = fˆ(ρ, φ),ρρû(R, φ) = û0 (φ) û(ρ, φ) = û(ρ, φ + 2π),0 6 ρ 6 R, 0 6 φ 6 2π(3)0 6 ρ 6 R, 0 6 φ 6 2πÏðåäïîëîæèì,÷òî:1.
Áóäåì ðàññìàòðèâàòü çàäà÷ó Äèðèõëå äëÿ óðàâíåíèÿ Ëàïëàñà2.u(x) ∈ C1 (D) ∩ C2 (D)3.u0 (x) ∈ C1 (Γ)Ïðåäïîëîæèì, ÷òîû0 (ρ, φ)ìîæíî ðàçëîæèòü â ðàâíîìåðíî ñõîäÿùèéñÿ ðÿä Ôóðüå, èÑäåëàåì ýòî:]a0 (ρ) ∑ [+ak (ρ) cos kφ + bk (ρ) sin kφ2∞û(ρ, φ) =k=1∫2π1û(ρ, ψ) cos kψdψ,ak (ρ) =π0∫2π1û(ρ, ψ) sin kψdψ,bk (ρ) =π035(k = 0, 1, . . . )(k = 0, 1, . . . )û(ρ, φ)òîæå.]A0 ∑ [+Ak cos kφ + Bk sin kφ2∞û0 (φ) =k=1∫2π1Ak =û0 (ψ) cos kψdψ, (k = 0, 1, . .
. )π0∫2π1Bk =û0 (ψ) sin kψdψ, (k = 0, 1, . . . )π0Ïîäñòàâèì∞ {[] ∑1 [ ′′1 ′1a′′a0 (ρ) + a′0 (ρ) +k (ρ) + ρ ak (ρ) −2ρk=1∞ [∑]1a0 (R) +ak (R) cos kφ + bk (R) sin kφ =2k=1}][]k2k21 ′′′b(ρ)−a(ρ)coskφ+b(ρ)+b(ρ)sinkφ=0kkkρ kρ2ρ2∞]A0 ∑ [+Ak (R) cos kφ + Bk (R) sin kφ2(4)k=1Ïðèðàâíèâàÿ êîýôôèöèåíòû ïðè ñîîòâåòñòâóþùèõ ñëàãàåìûõ ïîëó÷èì:1 ′k2(ρ)+a(ρ)−ak (ρ) = 0, a′′kρ kρ2 ak (R) = Ak , (k = 0, 1, . . .
)k21(ρ) + b′k (ρ) − 2 bk (ρ) = 0, b′′kρρ bk (R) = Bk (k = 0, 1, . . . )Íà ïåðâûé âçãëÿä íå õâàòàåò óñëîâèé. Íî ïðèρ→006ρ6R06ρ6Ríàøà ôóíêöèÿ→ ∞.À íàì áû õîòåëîñü ýòîãîèçáåæàòü.u- îãðàíè÷åííà.1|al (ρ)| 6π∫2π1|û(ρ, ϕ)|· | cos kψ|ψ 6 Mπ0Íåîáõîäèìî, ÷òîáûak∫2πdψ = 2M0áûëî îãðàíè÷åííî:Íà æàðãîíå ìîæíî ñêàçàòü, ÷òî|ak (0)|, |bk (0)| < ∞Ðåøåíèåì óðàâíåíèÿ Ýéëåðà áóäåò:ak (ρ) = αρµα[µ(µ − 1) + µ − k2 ]ρµ−2 ⇒ µ = ±k|{z}=0{ak (ρ) = C1k ρk + C2k ρ−ka0 (ρ) = c10 · 1 + c20 ln ρîáùåå ðåøåíèå óðàâíåíèÿ, ïðèk>0c2k = c20 = 0 ⇒ ak (ρ) = c1k ρk (k = 0, 1, . .
. )( ρ )kak (R) = c1k Rk = Ak ⇒ ak (ρ) = Ak, (k = 0, 1 . . . )R( ρ )kÀíàëîãè÷íî: bk (ρ) = Bk, (k = 1, 2 . . . )RÈç îãðàíè÷åííîñòè:Òî åñòü ìû ôîðìàëüíî ðåøèëè íàøó çàäà÷ó:∞( ρ )kA0 ∑û(ρ, φ) =+(Ak cos kφ + Bk sin kφ)R2(∗)k=1Ïåðåéä¼ì îáðàòíî ê äåêàðòîâûì êîîðèäèíàòàì:z = x1 + ix2 = ρeiφ = ρ cos φ + iρ sin φkk ikφkkz = ρ e = ρ cos kφ + iρ sin kφρk cos kφ = Re(zk ) = Re(x1 + ix2 )k = pk (x1 , x2 );pk , qk - ãàðìîíè÷åñêèå ôóíêöèèρk sin kφ = Im(zk ) = Im(x1 + ix2 )k = qk (x1 , x2 )BkA0 ∑ Ak+pk (x1 , x2 ) + k qk (x1 , x2 )k2RRk=1∞u(x1 , x2 ) =(∗∗)36u0 (x) ∈ C(Γ).Òåîðåìà 6.1: ÏóñòüÒîãäà:u(x) ∈ C∞ (D) ∩ C(D)ïðè÷¼ì ïðè |x| < R < R11)ñóùåñòâóåò åäèíñòâåííîå êëàññè÷åñêîå ðåøåíèå çàäà÷è Äèðèõëå2) Ýòî ðåøåíèå ïðåäñòàâèìî ñõîäÿùèìèñÿ ðÿäàìè (*) è (**),ðÿäû ñõîäÿòñÿàáñîëþòíî è ðàâíîìåðíî.3)Ëþáûå ÷àñòíûå ïðîèçâîäíûå ïîâ (*), (**) ïîä çíàêîì∑x1èx2ìîãóò áûòü íàéäåíû ñîîòâåòñòâóþùèì äèôôåðåíöèðîâàíèåì4) Ýòî ðåøåíèå ïðåäñòàâèìî ôîðìóëîé Ïóàññîíà:I 2212πRu(x) =R − |x|u0 (ξ)dSξ ,|x − ξ|2∀x ∈ DΓÅäèíñòâåííîñòü äîêàçûâàòü ïîêà íå óìååì.∫2π1)∃M1π: |u0 (x)| 6 M ⇒ |Ak | 6|û0 (ψ)| cos |kψ|dψ 6 2M;0àíàëîãè÷íî:|Bk | 6 2MA0 ∑ AkBkA0 ∑ AkBk++u(x1 , x2 ) =p(x,x)+q(x,x)=Re(zk ) + k Im(zk )1212kkkkk22RRRRk=1k=1∞∞∑Bk kA0 ∑ Ak k+z;W2 (z) =zW1 (z) =2RkRkk=1k=1( ) Ak k 2M kR1= 2Mqk ; 0 < q < 1 k z 6 k |z| = 2MR2RR∞∞Òî åñòü ìû ìàæîðèðîâàëè ðÿä ñõîäÿùåéñÿ ïðîãðåññèåé.
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