TY - JOUR
T1 - On a Free Boundary Problem for the Curvature Flow with Driving Force
AU - Guo, Jong Shenq
AU - Matano, Hiroshi
AU - Shimojo, Masahiko
AU - Wu, Chang-Hong
PY - 2016/3/1
Y1 - 2016/3/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84954367613&partnerID=8YFLogxK
U2 - 10.1007/s00205-015-0920-8
DO - 10.1007/s00205-015-0920-8
M3 - Article
AN - SCOPUS:84954367613
SN - 0003-9527
VL - 219
SP - 1207
EP - 1272
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -