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）︰2017/05/16
考試時限（分鐘）：160
試題 :
The gradients appeared in this paper are weak derivatives. You have to write
your calculations and reasonings clearly.
(1) (45 points) Assume 1 < p < ∞, and $U \subset \mathbb R^N$ is a bounded open
set.
(1) Prove that if $u \in W^{1, p}(U)$, then $|u| \in W^{1, p}(U)$.
(2) Prove $u \in W^{1, p}(U)$ implies $u^+, u^- \in W^{1, p}(U)$, and
\[Du^+ = \begin{cases}
Du & \text{a.e. on }\{u > 0\},\\
0 & \text{a.e. on }\{u \le 0\},
\end{cases}\]
\[Du^- = \begin{cases}
0 & \text{a.e. on }\{u > 0\},\\
-Du & \text{a.e. on }\{u \le 0\},
\end{cases}\]
(3) Prove that if $u \in W^{1, p}(U)$, then
Du = 0 a.e. on the set {u = 0}.
(2) (20 points) Let U be bounded, with a $C^1$ boundary. Show that a "typical"
function $u \in L^p(U)$ (1 ≦ p < ∞) does not have a trace on ∂U. More
precisely, prove there does not exist a bounded linear operator
\[T: L^p(U) \to L^p(\partial U)\]
such that $Tu = u|_{\partial U}$ where $u \in C(\overline U) \cap L^p(U)$.
(3) (45 points) Let $U \subset \mathbb R^N$ be a $C^2$, bounded open set. Sup-
pose $a^{ij}(x) \in C^1(\overline U)$ is uniformly strictly elliptic and $f
\in L^2(U)$. We define
\[Lu = -(a^{ij}(x)u_{x_i})_{x_j}.\]
(ⅰ) Prove that if $u \in H^1(U)$ is a weak solution to
Lu = f(x),
then $u \in H^2_\mathrm{loc}(U)$.
(ⅱ) Suppose 0 ∈ ∂U and in a neighborhood of 0, ∂U is a graph of the
function $x_N = \phi(x')$, where $x' = (x_1, x_2, \ldots, x_{N-1})$.
Define $y_i = x_i$ for i = 1, 2, 3, ..., N-1 and $y_N = x_N
- \phi(x')$. Suppose this coordinate change is valid for x ∈ D ∩ U,
where $D \subset \mathbb R^N$ is a smooth open neighborhood of 0. Let
W = y(D ∩ U). Show that $u(x) \in H^2(D \cap U)$ if and only if $v(y)
= u(x(y)) \in H^2(W)$.
(ⅲ) Prove that if $u \in H^1_0(U)$ is a weak solution to
Lu = f(x),
then $u \in H^2(U)$.