Óïðàæíåíèÿ ïî êóðñó ÏÎÒÎÊÈ ÑÐÅÄÍÅÉ ÊÐÈÂÈÇÍÛ 9 ñåìåñòð, V êóðñ, 2014 ãîä, ñïåöèàëüíîñòè "Ìàòåìàòèêà"  çà÷åò èäóò âñå óïðàæíåíèÿ, êîòîðûå âîçíèêàëè â ëåêöèÿõ, à òàêæå ñëåäóþùèå: ∂ 1. Ïóñòü ñåìåéñòâî êðèâûõ X : M × [0, ω) → R2 ýâîëþöèîíèðóåò ïî îáùåìó çàêîíó ∂t X(x, t) = W (x, t). Äîêàæèòå, ÷òî òîãäà äëèíà L(t) è îãðàíè÷èâàåìàÿ ïëîùàäü A(t) ýòîãî ñåìåéñòâà ýâîëþöèîíèðóþò ïî çàêîíàì Z Z dA dL =− hW, N i ds, =− hW, kN i ds. dt dt X(·,t) X(·,t) 2. Çàäàâ êðèâûå ÿâíî, ïîêàæèòå, ÷òî ðåøåíèåì óðàâåíåíèÿ ïîòîêà êðèâèçíû ÿâëÿþòñÿ ñëåäóþùèå íåÿâíî çàäàííûå êðèâûå: 2 (a) (paperclip solution ) ch(v(y − y0 )) = cos(v(x − x0 ))e−v t , |x| < π2 . 2 (b) (hairclip solution ) sh(v(y − y0 )) = cos(v(x − x0 ))e−v t , x ∈ R. 3. (ìàñøòàáèðîâàíèå âðåìåíè ) Äîêàæèòå, ÷òî åñëè X : M × [0, T ) → R2 ðåøåíèå óðàâíåíèÿ 2 2 ïîòîêà êðèâèçíû ∂X ∂t = −kN , òî ñåìåéñòâî êðèâûõ Xλ : M × [0, λ T ) → R , îïðåäåëÿåìûõ êàê Xλ (x, t) := λX(x, λ−2 t) áóäåò òàêæå ðåøåíèåì óðàâíåíèÿ ïîòîêà. 4. Ðåøåíèå X(x, t) : Mn × [0, ω) → Rn+1 óðàâíåíèÿ ïîòîêà ñðåäíåé êðèâèçíû (ÏÑÊ) ∂X ∂t = HN , X(x, 0) = X0 (x) íàçûâàåòñÿ ñàìîïîäîáíî ñæèìàþùèìñÿ (ÑÑ), åñëè ñóùåñòâóþò òàêàÿ òî÷êà x0 ∈ Rn+1 è ãëàäêàÿ ïîëîæèòåëüíàÿ ôóíêöèÿ λ(t) : [0, ω) → R+ , òàêèå, ÷òî X(x, t) = x0 + λ(t)(X0 (x) − x0 ) è limt→ω X(x, t) = x0 . (a) Äîêàæèòå, ÷òî åñëè X0 (x) : Mn → Rn+1 óäîâëåòâîðÿåò ñòðóêòóðíîìó óðàâíåíèþ H0 (x) + a hX0 (x) − x0 , N0 (x)i = 0 äëÿ íåêîòîðûõ a ∈ R, a > 0 è x0 ∈ Rn+1 , òî ñóùåñòâóåò ÑÑ ðåøåíèå X(x, t) ÏÑÊ òàêîå, ÷òî X(x, 0) = X0 (x). (b) Äîêàæèòå, ÷òî åñëè X(x, t) : Mn × [0, ω) → Rn+1 ÑÑ ê òî÷êå x0 ∈ Rn+1 ðåøåíèå ÏÑÊ, òî ëèáî H(x, t) ≡ 0, ëèáî X óäîâëåòâîðÿåò ñòðóêòóðíîìó óðàâíåíèþ H(x, t) + 1 hX(x, t) − x0 , N (x, t)i = 0. 2(ω − t) 5. Ðåøåíèå X(x, t) : Mn × [0, ω) → Rn+1 óðàâíåíèÿ ïîòîêà ñðåäíåé êðèâèçíû (ÏÑÊ) ∂X ∂t = HN , X(x, 0) = X0 (x) íàçûâàåòñÿ ñàìîïàðàëëåëüíî ñæèìàþùèìñÿ (ÑÏ), åñëè ñóùåñòâóåò òàêîé ãëàäêî çàâèñÿùèé îò âðåìåíè âåêòîð w(t) : [0, ω) → Rn+1 , òàêîé, ÷òî X(x, t) = X0 (x) + w(t). (a) Äîêàæèòå, ÷òî åñëè ãèïåðïîâåðõíîñòü X0 (x) : Mn → Rn+1 óäîâëåòâîðÿåò óðàâíåíèþ H0 (x) = hv, N0 (x)i äëÿ âñåõ x ∈ Mn è íåêîòîðîãî ïîñòîÿííîãî âåêòîðà v ∈ Rn+1 , òî îíà ãåíåðèðóåò ÑÏ ðåøåíèå óðàâíåíèÿ ÏÑÊ c w(t) = vt. (b) Äîêàæèòå, ÷òî åñëè X(x, t) : Mn × [0, ω) → Rn+1 ÑÏ ðåøåíèå óðàâíåíèÿ ÏÑÊ, òî ñóùåñòâóåò âåêòîð v ∈ Rn+1 òàêîé, ÷òî äëÿ âñåõ t ∈ [0, ω), x ∈ Mn ãèïåðïîâåðõíîñòè X(x, t) óäîâëåòâîðÿþò ñòðóêòóðíîìó óðàâíåíèþ H(x, t) = hv, N (x, t)i . 6. Ïóñòü A(t) è L(t) ïëîùàäü îãðàíè÷èâàåìîé îáëàñòè è äëèíà êðèâîé X(·, t). Îöåíèòå ïîâåäåíèå âî âðåìåíè èçîïåðèìåòðè÷åñêîé ðàçíîñòè L2 (t) − 4πA(t). Èñïîëüçóÿ òåîðåìó Ãðýéñîíà, âûâåäèòå îòñþäà èçîïåðèìåòðè÷åñêîå íåðàâåíñòâî äëÿ ãëàäêèõ çàìêíóòûõ âëîæåííûõ êðèâûõ. 7. Èñïîëüçóÿ òåîðåìó Ãðýéñîíà, äîêàæèòå, ÷òî ïëîùàäü, îãðàíè÷èâàåìàÿ êðèâûìè íîðìàëèçîâàííîãî ïîòîêà, Ã(t) ≡ π äëÿ ëþáîãî τ > 0. 8. Äîêàæèòå, ÷òî åñëè ñåìåéñòâî ãèïåðïîâåðõíîñòåé Xt (·) = X(·, t) óäîâëåòâîðÿåò óðàâíåíèþ ÏÑÊ è dµt ýëåìåíò ïëîùàäè Xt (·), òî ∂ dµt = −H 2 dµt . ∂t 9. Ïóñòü X∞ : Mn → Rn+1 ãèïåðïîâåðõíîñòü (apriori íå âëîæåííàÿ), óäîâëåòâîðÿþùàÿ ñòðóêòóðíîìó óðàâíåíèþ H + hX∞ , N i = 0. Èñïîëüçóÿ ñèëüíûé ïðèíöèï ìàêñèìóìà íà êîìïàêòíîì ìíîãîîáðàçèè Mn äîêàæèòå, ÷òî åñëè H ≡ const, òî |X∞ (x)| ≡ const, òî åñòü X∞ ñôåðà. Êàêîâ ðàäèóñ ýòîé ñôåðû? (Óêàçàíèå: Íàéäèòå ∆g(·,t) |X∞ |2 , ïîñëå ÷åãî îöåíèòå H èñïîëüçóÿ óðàâíåíèÿ äëÿ ∆H èç ëåêöèè)