Clamped biharmonic problem
= Clamped biharmonic problem
{title2=$\Delta^2u=f,\quad u=\partial_nu=0$}
The clamped problem uses the <weak solution> identity $\int\Delta u\Delta v=\int fv$ for every $v$ in the <clamped second-order Sobolev space>. For $f\in L^2(U)$, the <clamped Hessian identity> and the <Poincare inequality> make this a bounded <coercive bilinear form>. The <Lax-Milgram theorem> gives a unique <weak solution> without a mean-zero condition on $f$.