課程名稱︰偏微分方程式導論
課程性質︰系必修
課程教師︰黃信元
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012 5/15
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
2 __
A (20 pts.) Show that there exists at most one solution u \in C (Ω)∩C(Ω)
of the following problem
△u=f in Ω
u=g on ðΩ
1 n
where Ω is a C open bounded connected set \in R , f \in C(Ω) and g \in
C(ðΩ). (Hint: maximum principle or energy method.)
B (20 pts.) Solve the Dirichlet problem for the exterior of a circle.
2 2 2
u + u =0 for x +y >a
xx yy 2 2 2
u=h(θ) for x +y =a
2 2
u bounded as x +y ---> ∞
(You can use any method you wish.)
C (20 pts.) If u(x,y)=f(x/y) is a harmonic function, solve the ODE satisfied
by f.
D (20 pts.) Assume g is continuous function on ðB(0,r). Here B(0,r)={x||x|<r}.
The solution of
△u=0 in B(0,r)
u=g on ðB(0,r)
is given by 2 2
r -|x| g(y)
u(x)=－－－－ ∫ －－－－ dS(y).
ω r ðB(0,r) |x-y|^n
n
Show that
lim u(x)=g(x_0)
x->x_0
where x \in B(0,r), x_0 \in ðB(0,r).
E (20 pts.) State and prove Liouville's Theorem in PDE.
Note. \in 代表 "屬於"符號
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