[分析] PDE

[smartlwj (下次再努力)]

           n                                                     2  __
Let Ω in R  be open. Show that if there exists a function u ε C ( Ω )

vanishing on (boundary)Ω for which the quotient
                        2
                ∫ |▽u|
                 Ω
                ----------
                     2
                ∫  u
                 Ω

reaches its infimum λ, then u is an eigenfunction for the eigenvalue λ,

so that △u + λu = 0 in Ω.

上次那題已解決，謝謝幫忙的版友.

再一次麻煩各位，請問這題應該怎麼下手呢?

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推 doubleN     :let ∫|u|^2=1, then inf[∫|▽u|^2] = λ
→ doubleN     :=> inf [∫|▽u|^2 -λ] = inf [∫|▽u|^2 -λ∫u^2]
→ doubleN     :let L[u] = ∫|▽u|^2 -λ∫u^2
→ doubleN     :and vεC^∞ with compact support
→ doubleN     :then d/dt(L[u+tv]) = 0  when t=0 (local minimal)
→ doubleN     :=> ∫(-△u -λu)v = 0
→ doubleN     :=> -△u -λu = 0
→ smartlwj    :謝謝!! 我有個疑問~為什麼要用 u+tv 來做??怎麼想的
推 herstein    :高等微積分的作法....