Stability of stationary points for one-dimensional Willmore energy with spatially heterogeneous term

Abstract

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.