課程名稱︰偏微分方程導論 課程性質︰數學系必修 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04月19日，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.4 (4/19/2013) Solving the following problems. Please write down your computational of proof steps clearly on the answer sheets. A. Let I = (a,b) be an open interval in R. u : I → R is differentiable. 　 (a) (10 points) If u(x) has a local maximum occuring at c∈I, and u''(x) 　　　exists, prove that u''(c)≦0. 　 (b) (25 points) Assume that u∈C^2(I), and u''(x) + f(x)u'(x)≧0 for all 　　　 x∈I where f(x) is a bounded function defined in I. If there exists 　　　 c∈I such that u(x)≦u(c) for all x∈I, prove that u(x) is a constant 　　　 function. [Hint: If false, say u(d) < u(c) with c < d < b, show that 　　　 α > 0 and ε > 0 can be chosen so that w(x) = u(x) +ε(exp(α(x-c))-1) 　　　 satisfies w''(x) + f(x)w'(x) > 0 for a < x< d, and w(x) has an absolute 　　　 maximum occuring in (a,d).] 　 (c) (5 points) If u(x) satisfies u''(x) exp(u(x)) = - x for 0 < x < 1, prove 　　　 that u(x) cannot attain a minimum in (0,1). B. (25 points) Let λ0 < λ1 < λ2 < ... < λn < ... be the complete set of the 　 eigenvalue problem 　　　w'' + λw = 0 in 0 < x < L 　　　with boundary conditions w'(0)-a0*w(0) = w'(L) + aLw(L) = 0 , 　 where a0 > 0 and aL > 0 are constnats. 　 Prove that lim[λn - (nπ/L)^2] = 2*(a0 + aL)/L 　　　　　　　n->∞ 　　　　　　　　　　　 3　　　　　　　　　　　　　　　　　　　　　-2 C. (10 points) If u : R → R is harmonic, prove that u(x) = u(x*|x| )/|x| is 　 also harmonic. --

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