We study a free boundary problem associated with the curvature dependent
motion of planar curves in the upper half plane whose two endpoints slide along the
horizontal axis with prescribed fixed contact angles. Our first main result concerns
the classification of solutions; every solution falls into one of the three categories,
namely, area expanding, area bounded and area shrinking types. We then study in
detail the asymptotic behavior of solutions in each category. Among other things
we show that solutions are asymptotically self-similar both in the area expanding
and the area shrinking cases, while solutions converge to either a stationary solution
or a traveling wave in the area bounded case. We also prove results on the concavity
properties of solutions. One of the main tools of this paper is the intersection number
principle, however in order to deal with solutions with free boundaries, we introduce
what we call “the extended intersection number principle”, which turns out to be
exceedingly useful in handling curves with moving endpoints.
Relation:
Archive for Rational Mechanics and Analysis 219, pp.1207-1272
