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）︰2017/04/11 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) (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 $1 \le q < p^*$. -- 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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