TY - JOUR
T1 - Existence of planar curves minimizing length and curvature
JF - Proc. Steklov Inst. Math. 270 (2010) 43-56
Y1 - 2010
A1 - Ugo Boscain
A1 - GrĂ©goire Charlot
A1 - Francesco Rossi
AB - In 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.
PB - Springer
UR - http://hdl.handle.net/1963/4107
U1 - 297
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -