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）︰2017/04/11
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) (30 points) Assume U is an open bounded set in $\mathbb R^N$ with $C^1$
boundary ∂U. Suppose that $u \in W^{1, p}(U)$. Then
\[u \in W^{1, p}_0(U)\text{ if and only if }Tu = 0\text{ on }\partial U,\]
where T is the trace operator.
(2) (20 points) State and prove the Gagliardo-Nirenberg-Sobolev inequality for
functions $u \in C^1_0(\mathbb R^N)$. (You expect to obtain an inequality
between the $L^{p^*}$ norm of u and the $L^p$ norm of ▽u. You have to give
a relation to $p^*$ and p by scaling analysis.)
(3) (20 points) Prove the following Morrey's inequality: Assume N < p ≦ ∞.
Then there exists a constant C, dependeing only on p and N such that
\[\|u\|_{C^{0, \gamma}(\mathbb R^N)} \le C\|u\|_{W^{1, p}(\mathbb R^N)}\]
for all $u \in C^1(\mathbb R^N)$, where $\gamma := 1-\frac Np$.
(4) (30 points) Prove the Rellich-Kondrachov Compactness theorem: Assume U is a
bounded open set in $\mathbb R^N$ and ∂U is $C^1$. Suppose that 1 ≦ p < N.
Then
\[W^{1, p}(U) \subset \subset L^q(U)\]
for each $1 \le q < p^*$.