课程名称︰偏微分方程导论 课程性质︰数学系大三必修 课程教师︰林绍雄 开课学院：理学院 开课系所︰数学系 考试日期︰2013年05月13日(五)，08:10-09:10 考试时限：60分钟 是否需发放奖励金：是 试题 : 　　　　　　　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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