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

Michael Goldman, Matteo Novaga, Matthias Röger

TL;DR

This work analyzes a volume-constrained variational model that couples local perimeter and bending energies with a nonlocal Riesz interaction in dimensions two and three. The authors establish quantitative links between bending (elastica/Willmore) deficits and isoperimetric deficits in the planar setting, proving that for small charge the minimizers are balls or centered annuli, and balls remain optimal in 3D for small charge. They also derive phase diagrams and stability estimates for planar charged drops, showing transitions between annular and ball minimizers and nonexistence regimes at large charge; in 3D they obtain universal energy bounds, area/diameter estimates, and rigidity results that yield ball-minimization for small charge, while large-charge behavior remains more delicate with partial nonexistence results. Overall, the bending term regularizes the nonlocal problem, producing well-posedness in regimes and revealing rich geometry (balls vs. annuli) governing minimizers across dimensions and charge scales.

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.

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

TL;DR

This work analyzes a volume-constrained variational model that couples local perimeter and bending energies with a nonlocal Riesz interaction in dimensions two and three. The authors establish quantitative links between bending (elastica/Willmore) deficits and isoperimetric deficits in the planar setting, proving that for small charge the minimizers are balls or centered annuli, and balls remain optimal in 3D for small charge. They also derive phase diagrams and stability estimates for planar charged drops, showing transitions between annular and ball minimizers and nonexistence regimes at large charge; in 3D they obtain universal energy bounds, area/diameter estimates, and rigidity results that yield ball-minimization for small charge, while large-charge behavior remains more delicate with partial nonexistence results. Overall, the bending term regularizes the nonlocal problem, producing well-posedness in regimes and revealing rich geometry (balls vs. annuli) governing minimizers across dimensions and charge scales.

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.

Paper Structure

This paper contains 15 sections, 35 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The planar case: uncharged drops
  3. Simply connected sets: controlling the asymmetry index by the elastica deficit
  4. Minimizers of $\mathcal{F}_\lambda$
  5. Stability estimates
  6. The planar case: charged drops
  7. Minimization in the class of simply connected sets
  8. Minimization in the class $\mathcal{M}(|B_1|)$.
  9. Non-existence of minimizers for large charge
  10. The three-dimensional case
  11. Upper bound on the energy
  12. Area and diameter bounds
  13. Minimality of the ball for small charge
  14. Properties of minimizers for large charge
  15. A non-existence result

Key Result

Theorem 1.1

There exists a universal constant $c_0>0$ such that for every set $E\in\mathcal{M}_{sc}(|B_1|)$, where $E\Delta F$ denotes the symmetric difference of the sets $E$ and $F$. Furthermore, there exist $\delta_0>0$ and $c_1 >0$ such that if $W(E)\leq W(B_1)+\delta_0$, then

Figures (1)

  • Figure 1: The phase diagram.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 57 more