→ Vulpix :你應該乘以u 02/16 18:14
推 doubleN :1.∫|▽u|^2 = λ∫|u|^2 - a∫|u|^2 02/16 19:13
→ doubleN :a > 0, u =\= 0 => λ > 0 02/16 19:14
→ doubleN :2.條件夠嗎? 02/16 19:20
→ smartlwj :第二題條件就只有這樣耶 02/16 23:27
1. Let λ in R, a > 0, and u be a smooth function defined on a
2 2 2
neighborhood of D = {(x,y) in R | x + y ≦ 1} s.t.
[ △u + λu = 0 in x^2+y^2 < 1
[
[ @u / @n = -au on x^2+y^2 = 1 (@ : partial)
where n is the unit outward normal vector to (boundary)D.
Prove that if u is not identically zero in x^2+y^2 < 1,
then λ > 0.
2. Let f in C^1(R^n) and suppose that for each open ball B that
there exists a solution of the boundary value problem
-△u = f in B
@u / @n = 0 on (partial)B
where n is the outward unit normal vector field to (partial)B.
show that f = 0.
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關於第二題,問題的兩邊同時乘上f 然後再B上積分
可得 ∫f^2 = ∫-△u = ∫▽f‧▽u , 然後就不知道怎麼做下去了
目的是希望可以讓積分等於0, 然後就可以知道f就是0...
至於第一題的部分,我的作法也和第二題類似,做到後面就卡住了...T.T
請給我一點提示怎麼做, 謝謝
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