Uniqueness and minimality of Euler's elastica with monotone curvature

For an old problem of Euler’s elastica we prove the novel global property that every planar elastica with non-constant monotone curvature is uniquely minimal subject to the clamped boundary condition. We also partly extend this unique minimality to the length-penalised case; this result is new even in view of local minimality. As an application we prove uniqueness of global minimisers in the straightening problem for generic boundary angles.