01347nas a2200133 4500008004300000245006300043210006300106260001300169520093400182100001701116700002301133700002101156856003601177 2010 en_Ud 00aExistence of planar curves minimizing length and curvature0 aExistence of planar curves minimizing length and curvature bSpringer3 aIn this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\\\\int \\\\sqrt{1+K_\\\\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions.\\nFor this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.1 aBoscain, Ugo1 aCharlot, GrĂ©goire1 aRossi, Francesco uhttp://hdl.handle.net/1963/4107