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：具有抗弯能力的界面的弹性能量既依赖于其几何形状，也依赖于其材料组成。我们考虑平面上的这样一种异构界面，由一条曲线及其附加的密度函数共同建模。由此产生的能量捕捉了曲率与密度效应之间复杂的相互作用，类似于Canham-Helfrich泛函。我们通过倾角来描述该曲线，从而将平衡方程简化为一个二阶椭圆型系统。在简要的变分讨论之后，我们研究了相关的非局部$L^2$-梯度流演化问题——一个耦合的拟线性抛物型问题。我们分析了凸性、正定性及对称性等量的（非）保持性，以及系统的渐近行为。数值实验对结果进行了验证。