課程名稱︰偏微分方程一
課程性質︰數學系選修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期︰2007年01月17日
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
Choose 5 from the following 7 problems.
1.
Let U^+ = {x in |R^n : |x| < 1 , x_n > 0}.
Assume u in C^2(cl(U_+)), △u = 0 in U^+, and u = 0 on bd(U^+)∩{x_n = 0}.
Set
v(x) = u(x) if x_n ≧ 0
= u(x_1,...,x_(n-1),-x_n) if x_n < 0
for x in B_1 = {y in |R^n : |y| < 1}.
Prove v in C^2(B_1) and △v = 0
2.
Given a C^1 function g : [0,oo) -> |R with g(0) = 0. Show that
u(x,t) = [x/√(4π)]∫_[0,t] [(t-s)^(-3/2)] * e^[(-x^2)/(4t-4s)] * g(s) ds
is a solution of the problem
/ u_t - u_xx = 0 in |R_+ x (0,oo)
| u = 0 on |R_+ x {t = 0}
\ u = g on {x = 0} x [0,oo)
3.
Let u > 0 be a harmonic function in {x in |R^n : 0 < |x| < 1} with n ≧ 3.
(助教後來補上 sup{|u(x)| : |x| = 1 } < oo 的條件。)
(a) Prove that lim(x→0) u(x) = oo if limsup(x→0) u(x) = oo.
(b) Show that if lim(x→0) [|x|^(n-2)]u(x) = 0, then there is M > 0 such that
u(x) ≦ M (Hint : consider the solution a|x|^(2n+2) + b)
4.
Let B_r = {x in |R^n : |x| < r} and {u_k} be a sequence of harmonic functions
on B_1 satisfying sup{|u_k| : x in B_1} ≦ 2 for all k.
Show that there is a subsequence {u_(k_j)} and u in C^2(B_1) such that
u_(k_j) converges uniformly to u in B_(1/2) and △u = 0 in B_1.
5.
Suppose u in C^2 is a solution of u_tt - △u = 0 in |R^n x [0,oo).
Let
(B_(t_0 - t))(x_0) = {x in |R^n : |x - x_0| ≦ t_0 - t} for 0 ≦ t ≦ t_0
and
e(t) = (1/2)∫_(B_(t_0 - t))(x_0) [(u_t(x,t))^2 + |Du(x,t)|^2] dx.
Show that
(a) e'(t) ≦ 0 for 0 ≦ t ≦ t_0.
(b) u ≡ 0 in C = {(x,t) : 0 ≦ t ≦ t_0 , |x - x_0| ≦ t_0 - t} if
u ≡ u_t ≡ 0 on (B_(t_0))(x_0) x {t = 0}
6.
Let P(x) = P(x + L) be a continuous periodic function on |R with period
L > 0. Assume u is a bounded solution of
/ u_t - △u = 0 in |R x (0,oo)
\ u = p(x) on |R x {t=0}
Show that lim(t->oo) u(x,t) = (1/L)∫_[0,L] P(y) dy for each x.
7.
Let u in C^2(|R x [0,oo)) solve the wave equation in one dimension
/ u_tt - u_xx = 0 in |R x (0,oo)
\ u = g , u_t = h on |R x {t = 0}
Suppose g and h hace compact support. Prove
(a) ∫_[-oo,oo] [(u_t(x,t))^2 + (u_x(x,t))^2] dx is constant in t.
(b) ∫_[-oo,oo] (u_t(x,t))^2 dx = ∫_[-oo,oo] (u_x(x,t))^2 dx for large t.
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