# The shape of low energy configurations of a thin elastic sheet with a single disclination

## TL;DR

This paper improves the understanding of the energy scaling law for thin elastic sheets with a disclination and proves that near-minimizers converge to a radially symmetric conical shape as thickness approaches zero.

## Contribution

It refines the energy scaling law for such sheets and establishes convergence of minimizers to a conical shape in the thin limit.

## Key findings

- Improved the lower bound in the energy scaling law.
- Proved convergence of minimizers to a conical shape as thickness tends to zero.
- Demonstrated weak convergence in Sobolev spaces for the shape of the sheet.

## Abstract

We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness $h$ as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to leading order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces $W^{2,2}(B_1\setminus B_\rho;\mathbb{R}^3)$ for every $0<\rho<1$, as the thickness $h$ is sent to 0.