A Willmore-Helfrich $L^2$-flow of curves with natural boundary conditions

Abstract

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.