Singular perturbation by bending for an adhesive obstacle problem

TL;DR

This paper investigates a one-dimensional free boundary problem from materials science, analyzing how bending energy perturbations influence the singular limit of the energy using $ ext{Gamma}$-convergence, revealing boundary behavior akin to phase transition models.

Contribution

It introduces a singular perturbation analysis of an obstacle problem with bending energy, establishing a $ ext{Gamma}$-convergence framework for the energy's limit as bending effects diminish.

Findings

Derived the singular limit energy depending on free boundary surfaces.

Identified the impact of higher-order bending energy on free boundary singularities.

Connected the problem to phase transition models through boundary behavior analysis.

Abstract

A free boundary problem arising from materials science is studied in one-dimensional case. The problem studied here is an obstacle problem for the non-convex energy consisting of a bending energy, tension and an adhesion energy. If the bending energy, which is a higher order term, is deleted then "edge" singularities of the solutions (surfaces) may occur at the free boundary as Alt-Caffarelli type variational problems. The main result of this paper is to give a singular limit of the energy utilizing the notion of $\Gamma$-convergence, when the bending energy can be regarded as a perturbation. This singular limit energy only depends on the state of surfaces at the free boundary as seen in singular perturbations for phase transition models.