The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

翻译：抗弯曲界面的弹性能既依赖于其几何形状，也依赖于其材料组成。我们考虑平面中的此类异质界面，将其建模为曲线并附加密度函数。由此产生的能量刻画出曲率与密度效应间的复杂相互作用，类似于Canham-Helfrich泛函。我们通过倾角描述曲线，使平衡方程简化为二阶椭圆系统。在简要的变分讨论之后，我们研究了相关的非局部$L^2$-梯度流演化——一个耦合拟线性抛物问题。我们分析了凸性、正性与对称性等量的（非）保持特性，以及系统的渐近行为。数值实验验证了所得结果。