In this project, we are interested in the configurations of curve networks, whose elastic energy are at equilibria achieved by gradient flows. Simpler cases to start the study include tripod-like networks and tree-type networks. Our approach is to view networks as unions of curves joint at junction points by setting up certain boundary conditions at junction/boundary points, and then consider geometric gradient flow of networks. In fact , various of ``boundary conditions" at the junction points provide different type of difficulty for the geometric gradient flow. Among various boundary conditions, the most difficult one is the so-called clamped boundary condition, namely the end point is fixed and the tangent indicatrix at any end point is a prescribed (constant) unit vector. The main result in this case is stated in our report.
|Effective start/end date||2018/08/01 → 2019/10/31|
- second-order parabolic equation
- clamped boundary condition
- Willmore functional of curves
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