Deformation of framed curves with boundary conditions

We provide a general approach to deform framed curves while preserving their clamped boundary conditions (this includes closed framed curves) as well as properties of their curvatures. We apply this to director theories, which involve a curve $$\gamma : (0, 1)\rightarrow \mathbb {R}^3$$ γ : ( 0 , 1 ) → R 3 and orthonormal directors $$d_1$$ d 1 , $$d_2$$ d 2 , $$d_3: (0,1)\rightarrow \mathbb {S}^2$$ d 3 : ( 0 , 1 ) → S 2 with $$d_1 = \gamma '$$ d 1 = γ ′ . We show that $$\gamma $$ γ and the $$d_i$$ d i can be approximated smoothly while preserving clamped boundary conditions at both ends. The approximation process also preserves conditions of the form $$d_i\cdot d_j' = 0$$ d i · d j ′ = 0 . Moreover, it is continuous with respect to natural functionals on framed curves. In the context of $$\Gamma $$ Γ -convergence, our approach allows to construct recovery sequences for director theories with prescribed clamped boundary conditions. We provide one simple application of this kind. Finally, we use similar ideas to derive Euler–Lagrange equations for functionals on framed curves satisfying clamped boundary conditions.