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

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 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 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. 1 We choose to keep the factor 1 4 in dimension three to stick with the traditional notation.F λ,Q (E).(1.3)We start by considering the planar problem d = 2 and first focus on the uncharged case Q = 0. For λ = 0 no global minimizer exists in M(|B 1 |), but it has been recently shown in [7,14] that balls minimize the elastica energy under volume constraint in the class of simply connected sets. Our first result is a quantitative version of this fact in the spirit of the quantitative isoperimetric inequality [18,10].Theorem 1.1. There exists a universal constant c 0 > 0 such that for every set E ∈ M sc (|B 1 |),where E∆F denotes the symmetric difference of the sets E and F . Furthermore, there exist δ 0 > 0 and c 1 > 0 such that if W (E) ≤ W (B 1 ) + δ 0 , then W (E) − W (B 1 ) ≥ c 1 (P (E) − P (B 1 )).

Connected Coulomb columns: analysis and numerics

“…The most important limitation of our analysis is of course that the connected perimeter used here is strictly limited to the two-dimensional case. A possible avenue to pursue in higher-dimensional settings may be to add a Willmore-type term [GR17,GNR20]. The disconnectedness penalty for phase fields has been shown to preserve connectedness when combined with diffuse Willmore functionals [DLW17,DW18].…”

The average distance problem with perimeter-to-area ratio penalization

“…Also related are the papers by Paolini and Stepanov [22], Santambrogio and Tilli [23], Tilli [26], Lemenant and Mainini [19], Slepčev [25], and the review paper by Lemenant [18]. Similar variational problems entailing a competition between classical perimeter and nonlocal repulsive interaction were studied by Muratov and Knüpfer [21], Goldman, Novaga and Ruffini [16], and Goldman, Novaga and Röger [15]. Figalli, Fusco, Maggi, Millot, and Morrini studied a competition between a nonlocal s-perimeter and a nonlocal repulsive interaction term [13].…”

“…So in this case if the optimal constant in (2.8) is obtained by a circle, the optimal shape for (1.1) is a circle. An interesting question worthy further consideration is if the circle would be the minimizer for other parameters, as in similar discussions given in [21,16,15,13]. Another natural question is to ask if in general one may improve the C 1,1 regularity by combining the established results with elliptic regularity theory, given that the variation of the perimeter-to-area ratio leads to a system of second order differential equations of the boundary parametrization.…”

The average distance problem with an Euler elastica penalization

“…Meanwhile, if Σ is assumed to vary among sets Ω consisting of discrete points with a fixed cardinality, say k, then the minimization of the functional in (1.2), often named the quantization error in this case, is related to the centroidal Voronoi tessellations [11] and k-means, which are widely studied in subjects such as vector quantization, signal compression, sensor and resource placement, geometric meshing, and so on [12]. Similar variational problems entailing a competition between classical perimeter and nonlocal repulsive interaction were studied by Muratov and Knüpfer [21], Goldman, Novaga and Ruffini [17], and Goldman, Novaga and Röger [16]. Figalli, Fusco, Maggi, Millot, and Morrini studied a competition between a nonlocal s-perimeter and a nonlocal repulsive interaction term [14].…”