推 yusd24 :1. 用 max principle 09/04 23:29
→ yusd24 :C^2 = continuously differentiable. R^2 實數平面 09/04 23:30
→ yusd24 : ^ twice 09/04 23:31
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~
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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