※ 引述《gok338 (HuaHua)》之銘言：
: 1.試判斷正確性，並證明
: There exists no function U(x,y) which is C^2 in R^2 and satisfies
: U(x,y) = 0 on x^2 + xy + y^2 =1, U(x,y)≧0 and Uxx + Uyy = 1+U^2
: for x^2 + xy +y^2＜1.
: 還有C^2 in R^2 是什麼意思?? 謝謝~
C^2 二次可微且微分連續的函數 R^2 實數平面
Uxx + Uyy = 1+U^2 > 0
Claim: U can't attend its maximum value inside D.
First, D is a compact set in R^2 and U is a continuous function on it.
U must attend its maximum somewhere in D.
Suppose not, the maximum point is assumed at p in D. Since U is C^2,
Uxx(p)=<0, and Uyy(p)=<0. This contracts to the fact Uxx+Uyy=1+U^2 >0.
Now U=0 on boundary of D, so U<=0 in D by our claim.
Together with U>=0 we have U=0 on D. But if U=0, 0=Uxx+Uyy>1+U^2=1.
A contradiction.
So there is no such function U.
: 2.Assume that f_n: R→R (n=0.1.2...) is a sequence of differentiable
: functions s.t each f_n(x) is a solution of the equation
: y'(1 + x^2 + xy + y^2) = 1. If sup_n ∣f_n(1/n)∣＜∞,
: prove that there exists a subsequence f_n_k(x) s.t
: lim f_n_k(x) = f(x) exists for x in R.
: k→∞
: This limit f(x) must also be a solution of y'(1 + x^2 + xy + y^2) = 1.
: 謝謝~
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※ 編輯: yusd24 來自: 219.70.164.138 (09/04 23:44)