課程名稱︰偏微分方程式二
課程性質︰數學系選修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/03/21
考試時限（分鐘）：90
試題 :
The gradients appeared in this paper are weak derivatives. You have to write
your calculations and reasoning clearly.
(1) (20 points) Let p ≧ 1. Suppose that U is connected and $u \in W^{1, p}(U)$
satisfies
▽u = 0 a.e. in U.
Prove u is constant a.e. in U.
(2) (30 points) True or False. Let U = (0, 2). For each of the followings, you
have to prove or disprove the statement.
(a) Define
\[u(x) = \begin{cases}
x, & \text{if }0 < x \le 1,\\
x^2, & \text{if }1 \le x < 2.
\end{cases}\]
Is $u \in W^{1, 1}(U)$?
(b) Define
\[v(x) = \begin{cases}
1, & \text{if }0 < x \le 1,\\
0, & \text{if }1 \le x < 2.
\end{cases}\]
Is $v \in W^{1, 1}(U)$?
(3) (20 points) State the trace theorem.
(4) (30 points) Prove the trace theorem.