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

Part of: Existence theories Global differential geometry Magnetohydrodynamics and electrohydrodynamics

Published online by Cambridge University Press:  23 January 2019

Michael Goldman ,
Matteo Novaga  and
Matthias Röger
Michael Goldman
Affiliation:
Laboratoire Jacques-Louis Lions (CNRS, UMR 7598), Université Paris Diderot, F-75005Paris, France ([email protected])
Matteo Novaga
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127Pisa, Italy ([email protected])
Matthias Röger
Affiliation:
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany ([email protected])
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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’, that is, the weight of the Riesz interaction energy.

In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica 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 centred 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.

Keywords

Geometric variational problemscompeting interactionsnon-local perimeter perturbationWillmore functionalbending energyglobal minimizers

MSC classification

Primary: 49J20: Optimal control problems involving partial differential equations
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 76W99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 131 - 169
Copyright
Copyright © Royal Society of Edinburgh 2019

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