We consider a variational problem that consists of the Willmore energy of planar curves and a linear term in the curvature with a spatially heterogeneous coefficient. This coefficient is assumed to be a piecewise constant function. The variational problem arises from the modeling of the epidermal membrane in human skin. It has been predicted that the spatial heterogeneity of cell adhesion is one of the important factors in the protuberance formation of the membrane. We investigate the existence and stability of stationary points represented by symmetric graphs and find a condition for the existence of such stationary points by using the methods developed in the study of the Navier problem for Willmore energy. Especially, we give a necessary and sufficient condition for the existence of symmetric stationary points in terms of the gap size of the spatial heterogeneity. Moreover, we show some results on the stability of these stationary points.

我们考虑一个变分问题，该问题由平面曲线的Willmore能量和曲率中具有空间异质系数的线性项组成。假定该系数为分段常数函数。变异性问题源于人类皮肤中表皮膜的建模。已经预料到，细胞粘附的空间异质性是膜突起形成中的重要因素之一。我们研究对称图表示的固定点的存在性和稳定性，并使用研究威尔莫尔能量的Navier问题中开发的方法，找到此类固定点的存在条件。特别，根据空间异质性的间隙大小，我们为对称平稳点的存在提供了充要条件。此外，我们在这些固定点的稳定性上显示了一些结果。