Классические методы математической физики - 219 стр.

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n = (cos(n, x), cos(n, y), cos(n, t))
Γ
(x, t) = (x, y, t)
Γ cos
2
(n, t) = cos
2
(n, x) + cos
2
(n, y) = 1 /2
Z
Γ
1
cos(n, t)
(
∂u
∂x
cos(n, t) −
∂u
∂t
cos(n, x)
2
+
∂u
∂y
cos(n, t) −
∂u
∂t
cos(n, y)
2
)
dσ = 0.
cos(n, t) = 1/
√
2 Γ
∂u
∂x
cos(n, t) −
∂u
∂t
cos(n, x) = 0
Γ,
∂u
∂y
cos(n, t) −
∂u
∂t
cos(n, y) = 0
Γ.
∂u/∂x
cos(n, x)
=
∂u/∂y
cos(n, y)
=
∂u/∂t
cos(n, t)
Γ.
(x, t) ∈ Γ ∇u
n
λ = λ(x, t)
∇u(x, t) = λ(x, t)n(x, t) ∀(x, t) ∈ Γ.
l = l(x, t)
(x, t) ∈ Γ
∂u
∂l
(x, t) ≡ ∇u(x, t) · l = λ(x, t)n(x, t) · l(x, t) = 0.
n(x, t)
l(x, t)
u = const
t = 0 u = 0 u = 0
u = 0 (x
0
, t
0
)
u ∈ C
2
(R
3
+
)
(x
0
, t
0
) ∈ R
3
+
f
K(x
0
, t
0
) ϕ
0
ϕ
1
Σ = Σ(x
0
, t
0
) K(x
0
, t
0
) = {(x, t) : t > 0, t
0
− t > |x − x
0
|}
Σ(x
0
, t
0
) =
x ∈ R
2
: |x − x
0
| < at
0
f K
Çäåñü n = (cos(n, x), cos(n, y), cos(n, t)) - åäèíè÷íûé âåêòîð âíåøíåé íîð-
ìàëè ê ïîâåðõíîñòè Γ.
   Ëåãêî âèäåòü, ÷òî â êàæäîé òî÷êå (x, t) = (x, y, t) áîêîâîé ïîâåðõíîñòè
Γ âûïîëíÿþòñÿ óñëîâèÿ cos2 (n, t) = cos2 (n, x) + cos2 (n, y) = 1/2. Ñ ó÷åòîì
ýòîãî ðàâåíñòâî (5.8) ìîæíî ïåðåïèñàòü â âèäå
             (                            2                                2 )
      1         ∂u             ∂u                ∂u              ∂u
Z
                   cos(n, t) −    cos(n, x) +        cos(n, t) −    cos(n, y)      dσ = 0.
   cos(n, t)    ∂x             ∂t                 ∂y             ∂t
Γ
                        √                                                      (5.9)
Ïîñêîëüêó cos(n, t) = 1/ 2 íà Γ, òî èç (5.9) ñëåäóåò, ÷òî
 ∂u             ∂u                     ∂u             ∂u
    cos(n, t) −    cos(n, x) = 0 íà Γ,    cos(n, t) −    cos(n, y) = 0 íà Γ.
 ∂x             ∂t                     ∂y             ∂t
                                                                        (5.10)
Îòñþäà ïîëó÷àåì, ÷òî
                    ∂u/∂x     ∂u/∂y     ∂u/∂t
                            =         =          íà Γ.                        (5.11)
                   cos(n, x) cos(n, y) cos(n, t)
àâåíñòâà (5.11) îçíà÷àþò, ÷òî â êàæäîé òî÷êå (x, t) ∈ Γ âåêòîð ∇u è
âåêòîð íîðìàëè n ïàðàëëåëüíû, òàê ÷òî ñ íåêîòîðûì êîýèöèåíòîì ïðî-
ïîðöèîíàëüíîñòè λ = λ(x, t) ñïðàâåäëèâî ðàâåíñòâî
                      ∇u(x, t) = λ(x, t)n(x, t) ∀(x, t) ∈ Γ.                  (5.12)
  Îáîçíà÷èì ÷åðåç l = l(x, t) åäèíè÷íûé âåêòîð îáðàçóþùåé êîíóñà â
ïðîèçâîëüíîé òî÷êå (x, t) ∈ Γ. Ó÷èòûâàÿ (5.12), èìååì
              ∂u
                 (x, t) ≡ ∇u(x, t) · l = λ(x, t)n(x, t) · l(x, t) = 0.        (5.13)
              ∂l
Ïîñëåäíåå ðàâåíñòâî â (5.13) ñëåäóåò èç òîãî óñëîâèÿ, ÷òî âåêòîðû n(x, t) è
l(x, t) îðòîãîíàëüíû. àâåíñòâî (5.13) îçíà÷àåò, ÷òî íà êàæäîé îáðàçóþùåé
êîíóñà u = const. Íî âî âñåõ òî÷êàõ êîíóñà, ãäå îáðàçóþùèå ïåðåñåêàþò
ïëîñêîñòü t = 0, ñîãëàñíî (5.4) èìååì, ÷òî u = 0. Îòñþäà ñëåäóåò, ÷òî u = 0
âäîëü êàæäîé îáðàçóþùåé êîíóñà, à ñëåäîâàòåëüíî, u = 0 è â òî÷êå (x0 , t0 )
 .
   Ôàêòè÷åñêè ìû äîêàçàëè, ÷òî ðåøåíèå u ∈ C 2 (R3+ ) çàäà÷è Êîøè (5.1),
(5.2) ðàâíî íóëþ â ïðîèçâîëüíîé òî÷êå (x0 , t0 ) ∈ R3+ , åñëè ïðàâàÿ ÷àñòü f
óðàâíåíèÿ (5.1) ðàâíà íóëþ â êîíóñå K(x0 , t0 ), à óíêöèè ϕ0 è ϕ1 ðàâíû
íóëþ â êðóãåΣ = Σ(x0 , t0 ), ãäå K(x0 , t0 ) = {(x, t) : t > 0, t0 − t > |x − x0 |},
Σ(x0, t0) = x ∈ R2 : |x − x0| < at0 . Ïîä÷åðêíåì, ÷òî óêàçàííûé àêò
èìååò ìåñòî íåçàâèñèìî îò òîãî, êàêèå çíà÷åíèÿ (íóëåâûå èëè îòëè÷íûå
îò íóëÿ) ïðèíèìàåò ïðàâàÿ ÷àñòü f âíå çàìêíóòîãî êîíóñà K , à íà÷àëüíûå

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