| Abstract: |
| In this talk, we will discuss planar closed elastic curves with density-modulated stiffness, which were inspired by experiments on biological membranes. In 2023, Brazda et al. characterized the heterogeneous elastic curves as the critical points of a geometric functional defined as the sum of a generalized bending energy with density-modulated stiffness and a Dirichlet energy for the density, under constraints on the total length and the total mass. Since any elastica with constant density is a trivial critical point in the model of density-dependent elastic curves, the functional can be regarded as a generalization of the classical bending energy. On the other hand, if the coefficient in the generalized elastic energy is non-smooth, the existence of heterogeneous elastic curves with non-classical shapes can be anticipated. The purpose of this talk is to prove (i) the existence of infinitely many heterogeneous closed elastic curves with non-classical shapes, and (ii) the existence of weak gradient trajectories connecting the classical elastica to the heterogeneous elastic curves obtained in (i). This talk is based on a joint work with Professor Shinya Okabe of Tohoku University. |
|