= Clamped second-order Sobolev space
{title2=$H_0^2(U)$}
This is the closure of the compactly supported <test functions> $C_c^\infty(U)$ in the $H^2(U)$ <norm>. On a smooth bounded domain it consists precisely of the $H^2$ functions with zero value and zero <normal derivative> traces. Tangential first derivatives then vanish as well. The <clamped Hessian identity> makes the <Laplacian> <norm> equivalent to the full $H^2$ <norm> on this space, enabling a <Lax-Milgram theorem> treatment of the <clamped biharmonic problem>.
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