[分析] weak solution
3 @u 3
Let Lu = Σ a (x)------ , x=(x ,x ,x ) in Ω where Ω is an open set in R
k=1 k @x_k 1 2 3
∞ 2 2
and a (x) in C (Ω). Given f in L (Ω), we say that u is an L weak solution
k
2 ∞
of Lu = f in Ω if u in L (Ω) and <u, L'ψ>=<f,ψ> for all ψ in C (Ω)
loc c
3 @((a_k)u)
where L'u = -Σ -----------.
k=1 @x_k
Suppose that there is a constant c such that
∞
<f,φ>≦c∥L'φ∥ 2 , for all φ in C (Ω)
L (Ω) c
2
Please prove that there exist an L weak solution of Lu = f.
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請問這題要怎麼做?? 我不知道要怎麼下手
我想法大概是朝要想辦法去造出一序列的 u_k 然後証明收斂.
因為這個形狀看起來有點類似fixed point的樣子~可是我還是做不出來...
請給我一點題示..謝謝
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