Deformation of framed curves with boundary conditions

Abstract

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 and orthonormal directors d_1, d_2, d_3: (0,1)rightarrow mathbb {S}^2 with d_1 = gamma '. We show that gamma and the d_i can be approximated smoothly while preserving clamped boundary conditions at both ends. The approximation process also preserves conditions of the form d_icdot d_j' = 0. Moreover, it is continuous with respect to natural functionals on framed curves. In the context of Gamma -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.