[試題] 101下 林紹雄 偏微分方程導論 第六次小考

[t0444564 (艾利歐)]

課程名稱︰偏微分方程導論
課程性質︰數學系大三必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期︰2013年05月17日(五)，08:10-09:10
考試時限：60分鐘
是否需發放獎勵金：是

試題 :
　　　　　　Math 2206 (Introduction to PDE) Quiz No.6 (5/17/2013)

Solve the following problems. Please wirte down your computational or proof
clearly on the answer sheets.

A. (10 points) Let G = {(x,y)∈R^2 | x > 0, y > 0} be the first quadrant of the
　　plane. Write down the Green function for G with Dirichlet boundary
　　conditions.

B. (30 points) Suppose that u∈C^2 (B_(2a)(0)) is a nonnegative harmonic
　　function in B_(2a)(0)⊂R^n (where a > 0 is a constnat), prove that
　　　 a^(n-2)(a-|x|)　　　　　　　　　a^(n-2)(a+|x|)
　　　----------------u(0) ≦ u(x) ≦ ----------------u(0) for |x| < a.
　　　　(a+|x|)^(n-1)　　　　　　　　　 (a-|x|)^(n-1)

C. Determine which of the following statements is true. Each has 15 points.
　　(a) Let G = {(x,y)∈R^2 | y > 0}. If u∈C^2(G)∩C(bra(G)) is harmonic in G,
　　　　and u = 0 on ∂G, then u = 0 in G.
　　(b) Let G = {(x,y)∈R^2 | y≠ 0}. If u∈C^2(G)∩C(R^2) is harmonic in G,
　　　　then u∈C^2(R^2), and is harmonic in R^2.

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