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We study the evolution of curves with fixed length and clamped boundaryconditions 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 andparabolic smoothing of the solution. Applying previous results on long timeexistence and proving a constrained Lojasiewicz-Simon gradient inequality wefurthermore show convergence to a critical point as time tends to infinity.

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