The function (where is bounded with smooth boundary ) is a weak solution of the biharmonic equation

and

provided for all .

I want to show that if then the weak solution 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 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?