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

We consider regular open curves in $\mathbb{R}^n$ 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.

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Published 22 September 2014