Singular perturbation by bending for an adhesive obstacle problem

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.

24 pages, 8 figures

Cited by in corpus (9)

• Existence and concentration results for some fractional Schrödinger equations in ${\mathbb{R}}^N$ with magnetic fields
• Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field
• Elastic curves and phase transitions
• Multiplicity and concentration results for fractional Schrödinger-Poisson equations with magnetic fields and critical growth
• Overhanging of membranes and filaments adhering to periodic graph substrates
• Polar tangential angles and free elasticae
• Geometric inequalities involving mean curvature for closed surfaces
• Numerical analysis of elastica with obstacle and adhesion effects
• Some minimization problems for planar networks of elastic curve