Quantitative estimates for bending energies and applications to non-local variational problems

TL;DR

This paper analyzes a complex variational model combining perimeter, bending, and Riesz energies, revealing conditions under which minimizers are balls or annuli, and exploring the energy landscape's dependence on charge size.

Contribution

It provides new mathematical results on minimizers of a combined energy model, including existence, uniqueness, and bounds in 2D and 3D cases, with a focus on small charge regimes.

Findings

Minimizers are either balls or centered annuli for small charge in 2D.

Balls are the unique minimizers for small charge in 3D.

Non-existence of minimizers for large charge.

Abstract

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', i.e. the weight of the Riesz interaction energy. In the two-dimensional case we first prove that for simply connected sets of small elastic energy, the elastic deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centered annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.