= Lagrange identity for a fourth-order differential operator
{c}
For $L=D^4+q(x)$ where $q$ is a real <continuous function> on $[a,b]$, <integration by parts> gives
$$
\int_a^b\bigl(\overline u Lv-\overline{Lu}v\bigr)\,dx
=\left[\overline u v'''-\overline{u'}v''+\overline{u''}v'-\overline{u'''}v\right]_a^b.
$$
The potential terms cancel, and differentiating the boundary expression verifies the identity. Under <clamped boundary conditions> $u=u'=v=v'=0$ at both endpoints, this form vanishes. Conversely, requiring its vanishing against every clamped test <function> forces the adjoint <function> and its first <derivative> to vanish at both endpoints, because the second and third test-<function> <derivatives> are independent endpoint traces. Thus, on a regular finite <interval> with the usual fourth-order Sobolev domain, the clamped realization is <self-adjoint>. The equality of domains is part of this conclusion; formal <self-adjointness> of the expression alone does not supply it.
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