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

    I want to show that if $\displaystyle f\in L^2 (U)$ then the weak solution $\displaystyle u$ 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 $\displaystyle B[u,v]=\int_U \Delta u\Delta v dx$ 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; Dec 27th 2011 at 04:56 PM.
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