Диссертация (1103157), страница 14
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Ïðåäïîëîæèì, ÷òî îïåðàòîð Πq èíúåêòèrmâåí â LrI−c0 (Rk+1+ ), à îïåðàòîð Πφ èíúåêòèâåí â LI−d (R+ ) ïðè íåêîòîðûõ r ∈{1, 2, ∞}, c0 = (c, d1 + · · · + dm ) ∈ Rk+ × R1+ , d = (d1 , . . . , dm ) ∈ Rm+ . Ïîëîæèìkmqe(x, y) = q(x, φ(y)), x ∈ R+ , y ∈ R+ .
Òîãäà qe óäîâëåòâîðÿåò (1.8), (1.10) èîïåðàòîð Πqe èíúåêòèâåí â LrI−c00 (Rk+m ), ãäå c00 = (c, d).Äîêàçàòåëüñòâî ëåììû 2.5. Ïîëüçóÿñü ôîðìóëîé êîïëîùàäè (1.36), ìû ïîëó÷àåì ñëåäóþùåå ðàâåíñòâî:Z Zk+m(2π) 2 (M e−eq )(z, w) =xz−I y w−I exp(−q(x, φ(y)) dx dyRk+ Rm+Z Z ∞ZdSy=xz−I ts−1 exp(−q(x, t)) dx dty w−I,|∇φ(y)|Rk+ 0φ−1 (1)ãäå z ∈ Ck , Re z = c, w = (w1 , . . .
, wm ) ∈ Cm , Re w = d. Îòñþäà ñ ó÷¼òîìôîðìóëû (1.63) ïîëó÷àåòñÿ ôîðìóëà(M e−φ )(w) (M e−q )(z, s)(M e )(z, w) = (2π),Γ(s)−eq12s = w1 + · · · + w m .(2.33)Óòâåðæäåíèå ïðåäëîæåíèÿ ñëåäóåò èç ôîðìóëû (2.33) è òåîðåìû 2.2.Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 2.1. Ìîæíî âèäåòü, ÷òî ïðåäëîæåíèå 2.1 íåïîñðåäñòâåííî ñëåäóåò èç ïåðâîãî ïóíêòà òåîðåìû 2.4 è ëåììû 2.5.71Òåïåðü ìû ïåðåéä¼ì ê äîêàçàòåëüñòâó òåîðåìû 2.5.Äîêàçàòåëüñòâî òåîðåìû 2.5. Äëÿ ïðîñòîòû îáîçíà÷åíèé ìû áóäåì ïðåäïîëàααãàòü, ÷òî qj (x) = Cj (x1 j +· · ·+xnj )1/αj , j = 1, 2.  îáùåì ñëó÷àå äîêàçàòåëüñòâîàíàëîãè÷íî.Ïðèìåíÿÿ ôîðìóëó (1.39), ìû ïîëó÷àåì, ÷òîµ1 (Lq1 (p0 , p)) = µ2 (Lq2 (p0 , p)),p0 > 0, p ∈ Rn+ .Îòñþäà, ïîëüçóÿñü ëåììîé 1.4 ñ h(t) = exp(−tα1 ), ïðèõîäèì ê ñëåäóþùåìóðàâåíñòâó ïðè âñåõ p ∈ Rn+ :Zexp −C1α1 (p1 x1 )α1 + · · · + (pn xn )α1exp −C2α1 (p1 x1 )α2 + · · · + (pn xn )α2 αα1 µ1 (dx)Rn+Z=2µ2 (dx).Rn+Ñäåëàåì â ïåðâîì èíòåãðàëå çàìåíó ïåðåìåííûõ yj = xαj 1 , à âî âòîðîì yj =xαj 2 .
Îáîçíà÷èì ïîëó÷àþùèåñÿ ìåðû ÷åðåç µ01 è µ02 , ñîîòâåòñòâåííî. Êðîìå òîãî,ñäåëàåì çàìåíó ïàðàìåòðîâ uj = (C2 pj )α1 . Ìû ïîëó÷èì ðàâåíñòâîZexp(−Cuy) µ01 (dy)Rn+Z=1exp −(uγ y) γ µ02 (dy),u ∈ Rn+ ,(2.34)Rn+γãäå C = (C1 /C2 )α1 , γ = α2 /α1 , uγ = (u1 , . . . , uγn ).Çàìåòèì, ÷òî ëåâûé èíòåãðàë â ôîðìóëå (2.34) ÿâëÿåòñÿ âïîëíå ìîíîòîííîéôóíêöèåé u. Ñëåäîâàòåëüíî, ïðàâûé èíòåãðàë òàêæå ÿâëÿåòñÿ âïîëíå ìîíîòîííîé ôóíêöèåé.  ÷àñòíîñòè, âòîðàÿ ïðîèçâîäíàÿ ïî ïåðâîé ïàðå èíäåêñîâíåîòðèöàòåëüíà ïðè âñåõ u ∈ Rn+ :Z Xn Rn+k=11γuk yku1 + · · · + unγ γ1 −2 11γγγγy1 y2 (u y) + γ − 1 e−(u y) µ02 (dy) ≥ 0.(2.35)Ïîëîæèì u1 = · · · = un = t è çàìåòèì, ÷òî ïîäûíòåãðàëüíàÿ ôóíêöèÿ â ôîðìóëå (2.35) ìàæîðèðóåòñÿ ñâåðõó âûðàæåíèåì11n2γ−1 (y1 + · · · + yn ) γ t(y1 + · · · + yn ) γ + |γ| + 1 .72Òàêæå îòìåòèì, ÷òî óñëîâèå |x|2α1 ∈ L1 (Rn+ , µ2 ) âëå÷¼ò óñëîâèå (y1 +· · ·+yn )2/γ ∈L1 (Rn+ , µ02 ).
Ïîëüçóÿñü òåîðåìîé î ìàæîðèðóåìîé ñõîäèìîñòè, ïåðåéä¼ì â ôîðìóëå (2.35) ê ïðåäåëó ïðè t → +0. Ïîëó÷èì ñëåäóþùåå íåðàâåíñòâî:(γ − 1)n2γ−1Z1y1 y2 (y1 + · · · + yn ) γ −2 µ02 (dy) ≥ 0.Rn+Äàëåå çàìåòèì, ÷òî òàê êàê γ < 1, òî îòñþäà ñëåäóåò, ÷òî µ02 = 0 è µ2 = 0.Ñëåäîâàòåëüíî, Πq1 µ1 = 0. Ïî òåîðåìå 2.4 ïîëó÷èì, ÷òî µ1 = 0.
Òåîðåìà 2.5äîêàçàíà.Äëÿ äîêàçàòåëüñòâà ïðåäëîæåíèÿ 2.2 íàì ïîòðåáóþòñÿ îäíî âñïîìîãàòåëüíîå óòâåðæäåíèå. Íàïîìíèì, ÷òî ïðåîáðàçîâàíèå Ëàïëàñà ìåðû θ íà Rn+ çàäà¼òñÿ ôîðìóëîéZe−px θ(dx),(Lθ)(p) =p ∈ Rn+ .Rn+Äëÿ âñÿêîé íåîòðèöàòåëüíîé áîðåëåâñêîé ìåðû θ íà R2+ îïðåäåëèì ìåðó√√ïðàâèëîì θ(A) = θ( A) äëÿ âñÿêîãî áîðåëåâñêîãî ìíîæåñòâà A, ãäå√√θ√ √A = ( x1 , x2 ) ∈ R2+ | (x1 , x2 ) ∈ A .Ëåììà 2.6. (A) Ïóñòü ìåðà µ îïðåäåëåíà ôîðìóëîé (2.9). ÒîãäàZ∞exp −t −(Lµ)(p1 , p2 ) =√1t√p1 +√ dtp2,tp1 , p2 > 0.(2.36)0(B) Ïóñòü ìåðà ν îïðåäåëåíà ôîðìóëîé (2.10).
Òîãäà√(L ν)(p1 , p2 ) =Z∞√exp − t −0√1tp1 + p2 dt,tp1 , p2 > 0.(2.37)Äîêàçàòåëüñòâî ëåììû 2.6. Óòâåðæäåíèå (A). Äëÿ âñÿêîãî c > 0 ñïðàâåäëèâà ôîðìóëàZ∞2c2 dtK1 (2c) = exp −t −,ct t20ãäå ôóíêöèÿ K1 îïðåäåëåíà â ôîðìóëå (2.7). Ïîëàãàÿ c =√1 √x1 +x22 x1 x2è èñïîëüçóÿ73ôîðìóëó (2.9), ïîëó÷èì ñëåäóþùåå ðàâåíñòâî:µ(dx1 , dx2 ) =Z∞dx1 dx234π(x1 x2 ) 21dt1 1+.exp −t −4t x1 x2t2(2.38)a,−4u(2.39)0Îïðåäåëèì ôóíêöèþ√aea (u) = √ 3 exp2 πu 2a > 0, u > 0.Ïðåîáðàçîâàíèå Ëàïëàñà ôóíêöèè ea çàäà¼òñÿ ñëåäóþùåé ôîðìóëîé (ñì. [89, ñ.35, ôîðìóëà (28)]):√(Lea )(s) = e− as , s > 0.(2.40)Èç ôîðìóë (2.38) è (2.39), (2.40) ïðè a = t−1 ñëåäóåò ôîðìóëà (2.36).Óòâåðæäåíèå (B).
Ïóñòü A ⊂ R2+ ïðîèçâîëüíîå áîðåëåâñêîå ìíîæåñòâî.Îáîçíà÷èìAdiag = u ∈ R1+ | (u, u) ∈ A .Ñïðàâåäëèâà ñëåäóþùàÿ öåïî÷êà ðàâåíñòâ:√√ (2.10)ν(A) = ν( A) == 4Z√( A)diag√1 dv v= u=== 2exp − v2vZexp − u1 du.uAdiagÝòî ðàâåíñòâî ìîæíî ñôîðìóëèðîâàòü â ñëåäóþùåì âèäå:√1ν(dx1 , dx2 ) = 2x−1exp−1x1 δ(x1 − x2 ) dx1 dx2 .Ïîëüçóÿñü ýòîé ôîðìóëîé, ïîëó÷èì ñëåäóþùóþ öåïî÷êó ðàâåíñòâ, êîòîðàÿ äîêàçûâàåò ôîðìóëó (2.37):√(L ν)(p1 , p2 ) = 2Z∞e−s(p1 +p2 ) s−1 exp − 1s ds0√s−1 = tZ∞=====0√exp − t −√1tp1 + p2 dt.t74Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 2.2.
Ìû ïðèìåíèì òåîðåìó õàðàêòåðèçàöèè 1.4ê ôóíêöèè Π. Èç îïðåäåëåíèÿ (2.8) âèäíî, ÷òî ôóíêöèÿ Π óäîâëåòâîðÿåò ñâîéñòâó (3) òåîðåìû 1.4. Ïîâòîðíûì äèôôåðåíöèðîâàíèåì ôîðìóëû (2.8) ïî p0 ìûïîëó÷àåì ñëåäóþùåå ðàâåíñòâî:∂ 2Π1√√ √1(p,p)=exp−p+p2 ,01p0∂p20p0p0 > 0, p ∈ R2+ .(2.41)Èç ôîðìóëû (2.41) âèäíî, ÷òî ∂ 2 Π/∂p20 ≥ 0. Ñëåäîâàòåëüíî, âûïîëíåíî ñâîéñòâî(1) òåîðåìû 1.4.
Êðîìå òîãî, èç ôîðìóëû (2.41) ñëåäóåò, ÷òî2∂ 2Π−1 ∂ Π(λt, p) = λ(t, p),∂p20∂p20λ > 0, t > 0, p ∈ R2+ .Ñëåäîâàòåëüíî, Π óäîâëåòâîðÿåò ñâîéñòâó (2) òåîðåìû 1.4.Òåïåðü îòìåòèì, ó÷èòûâàÿ ôîðìóëû (2.36) è (2.37), ÷òî ñïðàâåäëèâû ðàâåí√ñòâà F1 = Lµ è F1/2 = L ν , ãäå ôóíêöèè F1 è F1/2 îïðåäåëåíû â ôîðìóëå (1.35).Ñëåäîâàòåëüíî, ôóíêöèè F1 è F1/2 âïîëíå ìîíîòîííû.
Òàêèì îáðàçîì, ôóíêöèÿΠ óäîâëåòâîðÿåò ñâîéñòâó (4) òåîðåìû 1.4 ïðè α = 1 è α = 1/2. Èç òåîðåìû1.4 ñ ó÷¼òîì ôîðìóëû (1.78) ñëåäóþò ïðåäñòàâëåíèÿ Π = Πq1 µ è Π = Πq1/2 ν .Ïðåäëîæåíèå 2.2 äîêàçàíî.7533.1Îáðàòíàÿ çàäà÷à ÄèðèõëåÍåéìàíà è å¼ïðèëîæåíèÿ â àêóñòè÷åñêîé òîìîãðàôèèÎñíîâíûå îïðåäåëåíèÿ è ïîñòàíîâêà çàäà÷ ýòîì ïàðàãðàôå ìû îïðåäåëèì îïåðàòîð ÄèðèõëåÍåéìàíà è ñôîðìóëèðóåìîáðàòíóþ çàäà÷ó ÄèðèõëåÍåéìàíà.
Ìû íå áóäåì ôîðìóëèðîâàòü òðåáîâàíèÿê ðåãóëÿðíîñòè âîçíèêàþùèõ ôóíêöèé è îáëàñòåé â îáùåì ñëó÷àå, òàê êàê ýòèòðåáîâàíèÿ áóäóò ðàçëè÷íûìè ïðè èçó÷åíèè ðàçëè÷íûõ âîïðîñîâ.Ïóñòü Mn (C) îáîçíà÷àåò ìíîæåñòâî êîìïëåêñíûõ ìàòðèö ðàçìåðà n×n. Ìûðàññìàòðèâàåì îïåðàòîðLA,V = −∆ − 2iAj (x) ∂x∂ j + V (x),(3.1)j=122∆=dX∂∂+ ··· + 2,2∂x1∂xnA = (A1 , . . . , Ad ),ãäå x ∈ D, êîýôôèöèåíòû A1 , . .
. , Ad , V äîñòàòî÷íî ðåãóëÿðíûå Mn (C)çíà÷íûå ôóíêöèè â D, à D ⊂ Rd (d ≥ 2) îãðàíè÷åííàÿ îáëàñòü ñ ãëàäêîéãðàíèöåé ∂D.Ìû ðàññìàòðèâàåì çàäà÷ó Äèðèõëå äëÿ îïåðàòîðà LA,V â D:LA,V ψ = Eψ â îáëàñòè D,(3.2a)ψ|∂D = f,(3.2b)ãäå f äîñòàòî÷íî ðåãóëÿðíàÿ ôóíêöèÿ íà ∂D, à E ∈ C ñïåêòðàëüíûéïàðàìåòð. Ìû ïðåäïîëàãàåì, ÷òîE íå ÿâëÿåòñÿ ñîáñòâåííûì çíà÷åíèåì çàäà÷è Äèðèõëåäëÿ îïåðàòîðà LA,V â D,(3.3)òàê ÷òî çàäà÷à (3.2a), (3.2b) èìååò åäèíñòâåííîå ðåøåíèå ψ â ïîäõîäÿùåì êëàññåôóíêöèé.Îïåðàòîð ÄèðèõëåÍåéìàíà ΛA,V = ΛA,V (E) ñîïîñòàâëÿåò ôóíêöèè f íà∂D ôóíêöèþ ΛA,V f íà ∂D, êîòîðàÿ îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîì:ΛA,V f =∂ψ∂ν+Xdj=1Aj νj f |∂D ,(3.4)76ãäå ψ ðåøåíèå çàäà÷è (3.2a), (3.2b), à ν åäèíè÷íûé âíåøíèé âåêòîð íîðìàëèê ∂D. ïîñëåäóþùèõ ïàðàãðàôàõ íàèáîëüøåå âíèìàíèå áóäåò óäåëÿòüñÿ ñëó÷àþn = 1.
 ýòîì ñëó÷àå, åñëè íå îãîâîðåíî ïðîòèâíîå, ìû ïðåäïîëàãàåì, ÷òîAj è V ïðèíàäëåæàò L∞ (D, C), çàäà÷à Äèðèõëå (3.2a)(3.2b) ðàññìàòðèâàåòñÿ ñ f ∈ H 1/2 (∂D), à ðåøåíèå ψ èùåòñÿ â êëàññå H 1 (D). Ïðè ýòîì îïåðàòîðÄèðèõëåÍåéìàíà ΛA,V îòîáðàæàåò ôóíêöèþ f ∈ H 1/2 (∂D) â ðàñïðåäåëåíèå−1/2ΛA,V f ∈ H −1/2 (∂D), îïðåäÿåìîå ïî ôîðìóëå (3.4), â êîòîðîé ∂ψ(∂D)∂ν |∂D ∈ Hîïðåäåëÿåòñÿ ñëåäóþùèì ñîîòíîøåíèåì: ∂ψ|,u=∂D∂νZ ∇ψ · ∇eu − 2ieuA · ψe + ue(V − E)ψ dx,(3.5)Dãäå u ∈ H 1/2 (∂D), à ue ïðîèçâîëüíàÿ ôóíêöèÿ êëàññà H 1 (D), óäîâëåòâîðÿþùàÿ ue|∂D = u. Çàìåòèì, ÷òî åñëè ψ óäîâëåòâîðÿåò (3.2a), òî îïðåäåëåíèå (3.5)íå çàâèñèò îò âûáîðà ïðîäîëæåíèÿ ue ôóíêöèè u.Ïóñòü òåïåðü g äîñòàòî÷íî ðåãóëÿðíàÿ GLn (C)-çíà÷íàÿ ôóíêöèÿ â D, òàêàÿ ÷òî g(x) = Idn , ãäå GLn (C) îáîçíà÷àåò ìíîæåñòâî îáðàòèìûõ êîìïëåêñíûõìàòðèö ðàçìåðà n × n, à Idn åäèíè÷íàÿ ìàòðèöà ðàçìåðà n × n. Ðàññìîòðèìñëåäóþùåå ïðåîáðàçîâàíèå êîýôôèöèåíòîâ:∂g −1Aj → Agj = gAj g −1 + ig ,∂xjj = 1, .
. . , d,(3.6a)dX∂g −1g−1−1V → V = gV g − g∆g − 2igAj,∂xj(3.6b)j=1ggè îáîçíà÷èì Ag = (A1 , . . . , Ad ). Ïîäñòàíîâêîé ïðîâåðÿåòñÿ, ÷òî ñïðàâåäëèâûôîðìóëûgLA,V g−1= −∆ − 2idXAgj (x) ∂x∂ j + V g (x),j=1ΛAg ,V g = ΛA,V ,ãäå g è g −1 ïîíèìàþòñÿ êàê îïåðàòîðû óìíîæåíèÿ íà ñîîòâåòñòâóþùóþ ôóíêöèþ. Òàêèì îáðàçîì, îïåðàòîð ΛA,V èíâàðèàíòåí îòíîñèòåëüíî ïðåîáðàçîâàíèé(3.6a), (3.6b), êîòîðûå ìû áóäåì íàçûâàòü êàëèáðîâî÷íûìè ïðåîáðàçîâàíèÿìè.77Îñíîâíàÿ çàäà÷à, êîòîðàÿ íàñ áóäåò èíòåðåñîâàòü, ôîðìóëèðóåòñÿ ñëåäóþùèìîáðàçîì.Çàäà÷à 3.1.
Ïóñòü çàäàí îïåðàòîð ΛA,V (E) ïðè ôèêñèðîâàííîì E (èëè ïðèE èç ôèêñèðîâàííîãî ìíîæåñòâà). Íàéòè A è V ïî ìîäóëþ êàëèáðîâî÷íûõïðåîáðàçîâàíèé (3.6a), (3.6b).Ìû áóäåì íàçûâàòü çàäà÷ó 3.1 îáðàòíîé çàäà÷åé ÄèðèõëåÍåéìàíà. Çàäà÷à 3.1 âîçíèêàåò ïðè ðàññìîòðåíèè ìíîãèõ âîïðîñîâ ìàòåìàòè÷åñêîé ôèçèêè.Îòìåòèì íåêîòîðûå èç íèõ.Ïðè n = 1 óðàâíåíèå (3.2a) ÿâëÿåòñÿ ìîäåëüíûì óðàâíåíèåì äëÿ ãàðìîíè÷åñêîãî ïî âðåìåíè (e−iωt ) àêóñòè÷åñêîãî äàâëåíèÿ â äâèæóùåéñÿ æèäêîñòè.Ïðè òàêîì ðàññìîòðåíèè E = 0,ωi ∇ρA = 2v +,c2 ρω2aωV = 2 + 2iω ,cc(3.7)ãäå c ñêîðîñòü çâóêà, ρ ïëîòíîñòü, v ñêîðîñòü æèäêîñòè, αω êîýôôèöèåíò ïîãëîùåíèÿ, à ω ôèêñèðîâàííàÿ ÷àñòîòà.
 ðàçëè÷íûõ ÷àñòíûõ ñëó÷àÿõòàêàÿ ìîäåëü ðàññìàòðèâàëàñü â ðàáîòàõ [7, 97, 40, 105, 67, 66]. Çàäà÷à 3.1 âýòîé ìîäåëè ïåðåôîðìóëèðóåòñÿ êàê çàäà÷à âîññòàíîâëåíèÿ ïàðàìåòðîâ æèäêîñòè c, ρ, v è αω ïî ãðàíè÷íûì èçìåðåíèÿì. Ýòà çàäà÷à èìååò ïðèëîæåíèÿ âìåäèöèíñêîé äèàãíîñòèêå. Ìû ðàññìàòðèâàåì ýòó çàäà÷ó áîëåå ïîäðîáíî â 3.2.Ïðè n ≥ 2 è d = 2 óðàâíåíèå (3.2a) âîçíèêàåò êàê âîëíîâîå óðàâíåíèå âìîäîâîì ïðåäñòàâëåíèè äëÿ ãàðìîíè÷åñêîãî ïî âðåìåíè (e−iωt ) àêóñòè÷åñêîãîäàâëåíèÿ â äâèæóùåéñÿ æèäêîñòè â òð¼õìåðíîì öèëèíäðå êîíå÷íîé âûñîòû èñ îñíîâàíèåì D.
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