课程名称︰偏微分方程式一
课程性质︰数学系选修
课程教师︰夏俊雄
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/11/08
考试时限（分钟）：170
试题 :
2
1. Suppose u is a bounded harmonic function defined on upper half space |R =
+
{(x , x )| x ∈ |R, x ≧ 0} such that u is continuous up to the boundary with
1 2 1 2
u(x , 0) = g(x ). Show that
1 1
(A) (8 %) If g ≡ 0, then u ≡ 0.
2
(B) (12 %) If g(x ) = exp(-x ), show that
1 1
-1
u(x , x ) = O(x ) as x → ∞, uniformly in x . (1)
1 2 2 2 1
2. Suppose Ω = (-π, π) and Ω = Ω ×(0, T]. The parabolic boundary of Ω
T T
＿＿
is defined as ∂Ω = Ω \Ω . Now, we consider the following equation
T T T
u = u (2)
t xx
supplemented with the initial-boundary condition
╭ u(x, 0) = x(π - |x|),
│
╯ u(-π, t) = 0, (3)
│
╰ u(π, t) = 0.
(A) (20 %) Now, you solve the equations (2)-(3) by the following Fourier series
scheme: Set
∞
u(x, t) = Σ a (t) sin nx, (4)
n=1 n
and evaluate each a (t) and show the convergence of (4) with the a (t) you
n n
obtain.
＿＿
(B) (20 %) Suppose that v(x, t) is continuous on Ω that satisfies (3) and for
T
each heat ball E(x, t; r) ⊂ Ω we have
T
2
1 |x-y|
v(x, t) = ──∬ v(y, s) ─── dyds. (5)
2 E(x, t; r) 2
4r (t-s)
Do you think v(x, t) is a solution to the equations (2)-(3)? Prove (make the
reasonings/proof) or disprove it (give an example).
N
3. (20 %) For fixed x ∈ |R , T ∈ |R, r > 0, we define the heat ball
2
N+1 N/2 |x-y| N
E(x, t; r) := {(y, s) ∈ |R | s ≦ t, [4π(t-s)] exp ─── ≦ r }. (6)
4(t-s)
Suppose the differentiable function u(x, t) satisfies the heat equation on some
N+1
Ω ⊂ |R and E(x, t; ρ) ⊂ Ω . We define
T T
2
1 |x-y|
f(r) = ──∬ u(y, s) ─── dyds. (7)
N E(x, t; r) 2
4r (t-s)
Show that f'(r) = 0 for 0 < r < ρ.
4.
(A) (10 %) State the maximum principle for the Cauchy problem
╭ u (x, t) = Δu(x, t),
╯ t (8)
╰ u(x, 0) = g(x),
∞ N N
where g(x) ∈ L (|R ) ∩ C(|R ).
(B) (20 %) Prove the maximum principle for the Cauchy problem by using the
maximum principle for the bounded domain.
5. (20 %) Solve the following differential equations:
╭ uu + 2u = 1,
│ x y
╯ (9)
│ 1
╰ u(x, x) = ─x.
2
╭ u - 4u = 0,
│ tt xx
│
╯ u (x, 0) = x, (10)
│ t
│ x
╰ u(x, 0) = e .