| Abstract: |
| The area-preserving curvature flow was first studied by Gage in 1986. He proved that any smooth closed planar curve evolving under this flow exists for all time and converges to a circle enclosing the same area. A natural question is how the dynamics change when the coefficients depend on the anisotropy. In this talk, we study the anisotropic area-preserving curvature flow for planar curves, where both the external forcing term and the coefficient of the curvature may exhibit anisotropy depending on the orientation of the curve. Under suitable assumptions on the anisotropic coefficients, we prove the uniqueness of traveling wave solutions. Moreover, we show that any closed convex curve evolving under the flow converges to this traveling wave solution. |
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