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

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative \(L^2\) $L^2$ -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 Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.