课程名称︰偏微分方程式二 课程性质︰数学系选修 课程教师︰夏俊雄 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/05/16 考试时限（分钟）：160 试题 : 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话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了 第03话 已经没什麽好期望了 第04话 被当、21都是存在的 第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊 第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋 第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题 第11话 最後留下的补考 第12话 我最爱的学分 --

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