A remark on elastic graphs with the symmetric cone obstacle

TL;DR

This paper investigates the variational problem for elastic energy on symmetric graphs with unilateral constraints, establishing uniqueness, regularity loss, and existence criteria based on obstacle size, and links solutions to dynamical equilibria.

Contribution

It provides new results on the uniqueness, regularity, and existence of minimizers for elastic graphs with symmetric cone obstacles, including a complete classification based on obstacle size.

Findings

Uniqueness of minimizers under symmetric cone constraints

Loss of regularity of minimizers

Existence and non-existence criteria depending on obstacle size

Abstract

This paper is concerned with the variational problem for the elastic energy defined on symmetric graphs under the unilateral constraint. Assuming that the obstacle function satisfies the symmetric cone condition, we prove (i) uniqueness of minimizers, (ii) loss of regularity of minimizers, and give (iii) complete classification of existence and non-existence of minimizers in terms of the size of obstacle. As an application, we characterize the solution of the obstacle problem as equilibrium of the corresponding dynamical problem.