= Vanishing boundary condition
A <vanishing boundary condition> sets a specified boundary quantity to zero. The quantity must be named: zero <boundary value> is a homogeneous <Dirichlet boundary condition>, while zero outward <normal derivative> is a homogeneous <Neumann boundary condition>. These are different restrictions.
Which boundary terms disappear under <integration by parts> depends on the operator and the permitted <test functions>. For a second-order weak problem, zero test-function trace eliminates $\int_{\partial U}(\partial_nu)v$. A fourth-order problem can also involve derivative traces, so a zero value alone does not remove every boundary term. On $[0,1]$, $u(x)=x(1-x)$ vanishes at both endpoints but has nonzero endpoint first <derivatives>; the <clamped boundary condition> would require those slopes to vanish as well.
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