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 $u \in H^2(U)$. -- 第01話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了 第03話 已經沒什麼好期望了 第04話 被當、21都是存在的 第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊 第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋 第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題 第11話 最後留下的補考 第12話 我最愛的學分 --

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