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）︰2017/03/09
試題 :
1. (20 points) Suppose $U \subset \mathbb R^N$ is an open bounded smooth domain
and η(x) ≧ 0 is a $C_0^\infty$ scalar function with $\int_{\mathbb R^N}
\eta(x)dx = 1$. If $f \in L^p(U)$ with 1 ≦ p < ∞, show that for each V⊂⊂U
\[|\eta_\epsilon*f - f|_{L^p(V)} \to 0,\]
as $\epsilon \to 0^+$, where $\eta_\epsilon(x) := \epsilon^{-N}\eta(\frac x
\epsilon)$.
2. (50 points) Show the following extension theorem: Let p ≧ 1. Suppose $U
\subset \mathbb R^N$ is an open bounded $C^1$ domain. For a given bounded
open set V such that U⊂⊂V, there exists a bounded linear operator
\[E: W^{1, p}(U) \to W^{1, p}(\mathbb R^N)\]
such that for each $u \in W^{1, p}(U)$:
(ⅰ) Eu = u almost everywhere in U,
(ⅱ) Eu has support within V, and
(ⅲ) \[\|Eu\|_{W^{1, p}(\mathbb R^N)} \le C\|u\|_{W^{1, p}(U)},\]
the constant C depending only on p, U and V.
3. (30 points) Redo 2. for the case that $U \subset \mathbb R^N$ is an open
bounded $C^2$ domain,
\[E: W^{2, p}(U) \to W^{2, p}(\mathbb R^N),\]
and replace 2. (ⅲ) by
\[\|Eu\|_{W^{2, p}(\mathbb R^N)} \le C\|u\|_{W^{2, p}(U)},\]
the constant C depending only on p, U and V.