Calculus of Variations and Geometric Measure Theory
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M. Goldman - M. Novaga - M. Röger

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

created by novaga on 02 Jan 2018
modified on 19 Feb 2020


Published Paper

Inserted: 2 jan 2018
Last Updated: 19 feb 2020

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Volume: 150
Number: 1
Pages: 131-169
Year: 2020


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.


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