The area-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be nonsimple. For highly symmetric convex curves, the flows converge to m-fold circles, while for Abresch–Langer type curves, the convergence to m-fold circles happens if and only if the enclosed algebraic area is positive.
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This work is partially supported by the National Natural Science Foundation of China 11101078, 11171064 and the Education Department Program of Liaoning Province L2010068.
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Wang, XL., Kong, LH. Area-preserving evolution of nonsimple symmetric plane curves. J. Evol. Equ. 14, 387–401 (2014). https://doi.org/10.1007/s00028-014-0219-5
Mathematics Subject Classification (2010)
- Primary 53C44
- Secondary 35K59
- Mean curvature flow
- nonlocal parabolic equation
- asymptotic behavior