課程名稱︰偏微分方程式一
課程性質︰必修
課程教師︰陳逸昆
開課學院：理學院
開課系所︰數學所
考試日期（年月日）︰2017/11/21
考試時限（分鐘）：100
試題 :
1. Derive an explicit solution formula to
u + 3 u + u = f in |R × (0,∞)
t x
u = g on |R × {0}.
3
2. Let G(x,y) be the Green's function for an open set Ω in |R for
Laplace's equation. Show that G is symmetric, i.e.,
G(x,y) = G(y,x)
for all x, y ∈ Ω.
3. Let u be the solution of
3
Δu = 0 in |R
+
3
u = g on ∂|R
+
given by Poisson's Formula for the half plane. Assume g is bounded and
3
g(x) = |x| for x ∈ ∂|R , |x| ≦ 1. Show that Du is not bounded.
+
4. Let
3
Ω = {x∈|R : |x| < 1 and |x-(1,0,0)| > 1}.
Construct a barrier function of Ω at origin O = (0,0,0).
Check that your answer satisfies the definition of the barrier function.