Existence and convergence of the length-preserving elastic flow of clamped curves

Abstract

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long time existence and proving a constrained Lojasiewicz-Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.