[試題] 101下 林紹雄 偏微分方程導論 第五次小考

[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13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.5(5/03/2013)

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

A. (30 points) Let G = {(x,y)∈|R^2 | x^2 + y^2 <1}.
　 Consider the boundary value problem:
　　　　　　　　　　　　　　　　　　　　　∂u
　　△u = 0 in G with boundary condition ----- + αu = 0 on ∂G,
　　　　　　　　　　　　　　　　　　　　　∂n
　 where n is the outer normal of ∂G, and α∈|R is a constant. Use separation
　 of variables to solve this problem, and show that this problem has multipe
　 solutions only when α≦0.

B. (30 points) Let G ⊂ |R^n be a bounded open domain whose boundary ∂G is
　 C^1 with n as its outer normal. Assume that u ∈C^2(G) ∩ C^1(bra(G))
　 satisfies Δu≧0 in G, and u(x_0) = max{u(x)|x∈bar(G)} at some x_0 ∈∂G.
　 If G satisfies the interior sphere condition at x_0, and u(x) is not a
　 constant, prove that (∂u/∂n)(x_0) > 0.

C. (10 points) Let G ⊂ |R^n be an open domain, and u∈C^2(G). Prove that u is
　 subharmonic in G iff Δu≧0 in G.

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