= Self-adjoint endpoint conditions for filament bending
For the <elastic filament> operator $K=A\partial_x^4$, two <integration by parts> operations give boundary form $A[\overline u v'''-\overline{u'}v''+\overline{u''}v'-\overline{u'''}v]$. Four elementary homogeneous choices at each endpoint are $h=h'=0$ (clamped), $h=h''=0$ (hinged), $h''=h'''=0$ (free), and $h'=h'''=0$ (slope-constrained and transverse-force-free). Each defines a <self-adjoint operator> on a finite interval when imposed at both ends. More general real <Robin boundary conditions> also give <self-adjointness>; the four choices are not an exhaustive classification of all boundary subspaces.
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