課程名稱︰偏微分方程式二
課程性質︰數學研究所基礎課
課程教師︰林太家
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期︰2015年03月31日(二)，10:20-12:10
考試時限：110分鐘
試題 :
Test 1 03/31/2015
1. 30%
Let f(x) = |x| for x∈I=(-1,1).
1 1,2
i. Prove that f∈H (I) = W (I). (10%)
2 2,2
ii. Is f∈H(I) = W (I)? Justify your answer. (10%)
2
d f
iii. Calculate ----- in weak sense. (10%)
2
dx
2. 30% 3
Let g(x) = log|x| for x=(x1,x2,x3)∈B={x∈R : |x|<1}, where
2 2 2
|x| = √(x1 + x2 + x3).
1 1,2
i. Prove that g∈H (I) = W (I). (10%)
ii. Find a sequence of functions {gk} such taht
∞ 1
gk∈C (B) for k∈N, and gk→g in H (B). (10%)
0
2 2,2
iii. Is g∈H (I) = W (I)? Justify your answer. (10%)
3. 20%
Assume 0 < β < γ≦1. Prove the interpolation inequality
(1-γ)/(1-β) (γ-β)/(1-β)
∥u∥0,γ ≦ (∥u∥0,β ) (∥u∥0,1 ).
C (U) C (U) C (U)
4. 20% 1,p
Prove that if n=1 and u∈W (U) for some 1≦p＜∞, then u is equal a.e. to
an absolutely continuous function, and u' (which exists a.e.) belongs to
p
L (0,1).
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