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We consider regular open curves in R n with fixed boundary points and moving according to the L 2 -gradient flow for a generalization of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely, at the boundary the curvature vector is equal to the normal projection of a fixed given vector. A long-time existence result together with subconvergence to critical points is proven.doi:10.4310/cag.2014.v22.n4.a2 fatcat:3ia6maxqg5cehftlpzadpx44yy