[分析] weak solution

[smartlwj (下次再努力)]

           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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