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Math Help - The biharmonic problem with Dirichlet and Neumann conditions concerning unique sol

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    The biharmonic problem with Dirichlet and Neumann conditions concerning unique sol

    The function  u \in H_0^2(U) (where U\subset \mathbb{R}^n is bounded with smooth boundary \partial U) is a weak solution of the biharmonic equation
    \Delta(\Delta u)=f  \mbox{ in }U
    and
    u=\frac{\partial u}{\partial\nu}=0\mbox{ on }\partial U
    provided \int_U \Delta u\Delta v dx=\int_U f v dx for all v\in H_0^2(U).

    I want to show that if f\in L^2 (U) then the weak solution u is unique. I imagine this problem satisfies the conditions of the Lax Milgram theorem when the integral is considered as a bilinear operator. However, I am wondering why one can't show that B[u,v]=\int_U \Delta u\Delta v dx defines an inner product on the $H_0^2(U)$ and apply the Riesz Representation Theorem directly to the problem? Am I missing something simple here?
    Last edited by paulk; December 27th 2011 at 05:56 PM.
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