Упражнения для самостоятельного решения

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)