 Clamped second-order Sobolev space

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.