Quantitative estimates for bending energies and applications to non-local variational problems
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.
Key words and phrases:Geometric variational problems, competing interactions, non-local perimeter perturbation, Willmore functional, bending energy, global minimizers
2000 Mathematics Subject Classification:49J20,53C21,49Q45,76W99
- 1 Introduction
- 2 The planar case: uncharged drops
- 3 The planar case: charged drops
- 4 The three-dimensional case
In recent years there has been a strong interest in variational models involving a competition between a perimeter type energy and a repulsive term of long-range nature (see for instance the recent review papers [8, 22] and the detailed discussion below). The aim of this paper is to start investigating the effects for this class of problems of higher order interfacial energies such as the Euler elastica in dimension two or the Willmore energy in dimension three. We will consider the simplest possible setting and study volume constrained minimization of functionals defined for sets with as
The different contributions are given by
the perimeter , defined as
the elastica or Willmore energy , defined as
where denotes the mean curvature of i.e. the curvature in dimension two and the sum of the principal curvatures in dimension three111We choose to keep the factor in dimension three to stick with the traditional notation.,
the Riesz interaction energy , defined for as
For functional (1.1) is arguably the simplest example of an isoperimetric type problem showing competition between a local attractive term with a non-local repulsive term.
In the case of Coulombic interactions, that is and , this model appears in a variety of contexts. It is for instance the celebrated Gamow liquid drop model for atomic nuclei  or
the sharp interface limit of the so-called Ohta-Kawasaki model for diblock copolymers [42, 3]. See also  for another application of this model.
Even though the picture is not complete, it has been shown that minimizers are balls for small [28, 29, 12, 25, 15] (actually they are the only stable critical sets ) and that no minimizers exist for large [28, 29, 37] (see also  for a simple proof of non-existence in the three dimensional case).
Many more questions related to pattern formation have been investigated for very closely related models, see for instance
[2, 1, 9, 46, 5, 23, 20, 30] for a non-exhaustive list. Other examples of functionals presenting this type of competition can be found for instance in shape memory alloys [31, 27],
micromagnetics  or epitaxial growth [4, 35]. However, the closest model is probably the one for charged liquid drops introduced in  where the Riesz interaction energy is
replaced by a capacitary term. Surprisingly, it has been shown in  that independently of the charge, no minimizers exist for this model (see also [40, 41]). It has been suggested
in  that a regularization by a Willmore type energy such as the one considered here might restore well-posedness. This paper can be seen as a first step in this direction.
The energy (1.1) could be seen as a toy model for charged elastic vesicles, where the Willmore energy represents a prototype of more general bending energies for fluid membranes and a Coulomb self-interactions refers to the energy of a uniformly charged body. Associated with this interpretation, we refer to the parameter regimes and by the terms small and large ‘charge’.
Our goal is to understand how the picture changes for (1.1) in the presence of a bending energy i.e. for . For and fixed volume, since an annulus of large radius has arbitrary small elastica energy and also arbitrary small Riesz interaction energy, one needs to either restrict the class of competitors to simply connected sets or to include the perimeter penalization (that is take ). In contrast, for , the Willmore functional is scaling invariant and is globally minimized by balls . It seems then natural to consider (1.1) for and study the stability of the ball. Let us point out that compared to the planar case, configurations with catenoid type parts allow for a much larger variety of structures with low energy. This makes the identification of optimal structures and the distinction between existence and non-existence of minimizers even more challenging.
Setting and main results
Let us set some notation and give our main results. We will always assume that the energy (1.1) contains the bending term and we set without loss of generality . For , we define
Regarding the volume constraint, as will be better explained later on, up to a rescaling there is no loss of generality in assuming that . For , we define the following classes of admissible sets
and consider the variational problems
We start by considering the planar problem and first focus on the uncharged case . For no global minimizer exists in , 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].
There exists a universal constant such that for every set ,
where denotes the symmetric difference of the sets and .
Furthermore, there exist and such that if , then
The proof is based on the idea of  for the proof of the quantitative isoperimetric inequality to reduce by a contradiction argument to the case of nearly spherical sets and then compute a Taylor expansion along the lines of .
As opposed to  which is based on an improved convergence theorem, we obtain the strong convergence to the ball directly from the energy and a delicate refinement of  (see Lemma 2.5).
Still in the case , we then remove the constraint on the sets to be simply connected but consider the minimization problem (1.3) for .
Let and . There exists such that for , minimizers of (1.3) are annuli while for they are balls.
Next, we turn to the stability estimates analogous to Theorem 1.1.
Let and be given by Theorem 1.2. Then, there exists a universal constant , such that for any and
while for any there exists a constant such that for any
where the minimum is taken among all sets minimizing in (which are either balls or annuli depending on ).
Let . There exists such that for and all , balls are the only minimizers of (1.2).
The first part of the theorem is a direct consequence of the minimality of the ball for and for for small enough.
The second point regarding the minimality of centered annuli is the most delicate part of the theorem. It requires first to argue that sets of small energy are almost annuli and
then to use the stability of annuli. The last part of the theorem regarding non-existence is obtained by noticing that if a minimizer exists then we can obtain a lower bound on the energy which is not compatible for large
with an upper bound obtained by constructing a suitable competitor.
We conclude the paper by studying the three dimensional case where a characterization of the energy landscape is even more difficult. As already pointed out, the Willmore energy is invariant under rescaling and is globally minimized by balls [52, 49]. We can thus focus on the case where we have competition between the Willmore energy and the Riesz interaction energy. Stability estimates for the Willmore energy have been obtained by De Lellis and Müller . Building on these, on the control of the isoperimetric deficit by the Willmore deficit obtained in  and a bound on the perimeter (see Proposition 4.3), we obtain that balls are minimizers of (1.3) for small .
For and , there exists such that for every , the only minimizers of (1.3) are balls.
Of course, since balls are also minimizers of the isoperimetric problem, a direct consequence of Theorem 1.6 is the minimality of the balls for (1.3) for every and every . For the case we are not able to prove or disprove a non-existence regime in the parameter space. Still, we show that if a minimizer exists for every then its isoperimetric quotient must degenerate as (see Proposition 4.10). This is somewhat reminiscent of earlier results obtained by Schygulla .
Finally, we obtain a non-existence result in the case and in the regime of sufficiently large charge.
For every , there exists such that for every with , no minimizer of in exists.
Outline and notation
In Section 2 we first consider the planar case in the absence of charge () before considering the case in Section 3. In the last section we finally consider the three dimensional case. For the reader’s convenience, the main theorems given in the introduction are restated in the respective sections and some of them have been extended by more detailed statements. Theorem 1.1, Theorem 1.2, and Theorem 1.4 correspond to Theorem 2.3, Theorem 2.7 and Theorem 2.10, respectively. The statements in Theorem 1.4 are collected from Proposition 3.4, Theorem 3.8, Proposition 3.9 and Proposition 3.14. Theorem 1.6 corresponds to Theorem 4.6, and Proposition 1.7 to Proposition 4.11.
For two real numbers the notation means that for some that is universal (unless dependencies are explicitly stated). Correspondingly we use the notation and write if and .
2. The planar case: uncharged drops
We start by investigating the planar case in the absence of charges i.e. . Our aim is both to characterize the minimizers and to show that the energy controls the distance to these minimizers.
2.1. Simply connected sets: controlling the asymmetry index by the elastica deficit
We first restrict ourselves to simply connected sets. By [7, 14] balls are the unique minimizers of the elastica energy among simply connected sets with prescribed volume. Since they also minimize the perimeter, balls are the unique minimizers of in this class. Using the quantitative isoperimetric inequality , one could directly obtain a quantitative inequality for which would however degenerate as . Our aim here is to show that actually the elastica energy itself controls the distance to balls. This is a quantitative version of [7, 14] which could be of independent interest.
Inspired by a strategy first used in [1, 10] (see for instance also [6, 11, 21] for a few other applications) which was building on ideas from , we first restrict ourselves to nearly spherical sets. More precisely, we consider sets such that is a graph over for some , i.e.
with . We will need the following estimate on the elastica energy of nearly spherical sets.
Let and be a nearly spherical set. Then,
Moreover, if and the barycenter of is equal to zero, then
Let us now compute the Taylor expansion of the integrand. Keeping only up to quadratic terms, we get that on the one hand,
and on the other hand,
Using that , we obtain (2.2).
If now , using that , we have
while if the barycenter is zero, using that ,
and thus for every ,
Since by Parseval’s identity,
and since for , the polynomial is always non-negative and vanishes only for , we have for all with sufficiently small
concluding the proof of (2.3). ∎
We also recall the following Taylor expansion of the perimeter for nearly spherical sets (see ).
Let and let be a nearly spherical set with represented as in (2.1), then
There exists a universal constant such that for every and every set ,
Furthermore, there exist and such that if , then
In order to prove this theorem we will need some auxiliary results. Even for simply connected sets, uniform bounds on the volume and the elastica energy are in general not sufficient to obtain a perimeter or a diameter control (see for example [7, Figure 1]). However, this is the case for sets with elastica energy sufficiently small.
There exist and such that for every with , there holds
We first prove, by contradiction, a corresponding bound for the diameter, and then deduce the perimeter bound.
Assume that there exists a sequence in with
In the following steps, the implicit constants in and estimates may depend on the fixed sequence but are independent of .
First, up to a rotation, we may assume that for every the vertical section is not empty. By Fubini’s Theorem, there exists such that . Then, there also exist and such that and .
Choose an oriented tangent field on . We claim that, for some independent of , we have
Indeed, assume that (2.11) does not hold so that there exists with
Without loss of generality, using another translation and rotation, we can assume that and (see Figure 2).
By the bound on the elastica energy we can locally write the part of containing as a graph of the form with and uniformly bounded slope (see [7, Lemma 2.1]). Moreover, , since for
and similarly for . Let be an arclength parametrization of . We may assume that with . By (2.12), . Let us assume for definiteness that (the other case being analogous). We then have by the bound on the elastica energy that for every ,
Let be a small constant chosen so that for ,
This implies that for , stays inside the cylinder . Furthermore,
Therefore, the graph splits the cylinder into two connected components with in one of the components and in the other (see Figure 2).
Hence, intersects the graph of which contradicts the fact that can be locally written as a graph. We have thus shown that (2.11) holds.
We recall that we assume (without loss of generality) that and . By the bound on the elastica energy and (2.11), for small enough (but not depending on ), and belong to two different connected components of , where . Let , respectively be the connected component containing , respectively . By [7, Lemma 2.1] and (2.11), we have and where without loss of generality, we can assume that for (see Figure 3). Let now be a non-negative bump function with and let be such that
If we replace in the component by , we obtain a new set with (since by construction ) and
where we have used that thanks to the energy estimate, is uniformly small in to make the Taylor expansion. The set is made of two drops and with mass and satisfying . From [7, Theorem 3.5], for every couple of drops and , with ,
for some . By scaling, we deduce that if we choose a ball such that , then
from which we obtain that
contradicting the fact that .
By the previous steps we know that there exists such that for all with . Therefore, after translation . We now choose an arclength parametrization , and obtain as in [7, Lemma 2.5]
from which the perimeter bound follows.
Let be a sequence in with as . For every let be a constant speed parametrization of . Then, after translation converges strongly in to a (unit speed) arclength parametrization of .
Consider the sets . It follows from  and our assumptions that is a minimizing sequence of the functional on . Moreover, has uniformly bounded perimeter by Lemma 2.4. For such sequences it is proved in [7, Section 4] that, after translation, must converge to . Hence converges to . This gives first weak convergence in which combined with the convergence of the energy gives the strong convergence. ∎
Proof of Theorem 2.3.
Without loss of generality, we may assume that and .
Assume for the sake of contradiction that (2.8) does not hold. Then, there exists a sequence such that
Since is bounded, this implies that as . We then obtain from Lemma 2.4 that the diameter of remains bounded. Hence, by Lemma 2.5 must converge up to translation to strongly in and thus by Sobolev embedding, also in for every . Thus, for large enough, is a graph over and is nearly spherical. Since the barycenter of is converging to zero, is also a graph over the ball centered in its barycenter. Up to a translation, this means that we can apply (2.3) and obtain that
which contradicts (2.13).
which is in contradiction with (2.15).
2.2. Minimizers of
We move on and remove the constraint that sets are simply connected. Since annuli of very large diameter have vanishing elastica energy, in order to have a well-posed problem, we need to consider . Let us recall that the energy is given by
Up to a rescaling, we may restrict ourselves to the problem
We will show below that depending on the value of , minimizers are either balls or annuli. Before stating the precise result we compute the energy of an annulus.
For every , the energy of an annulus with inner radius and area equal to is given by
The function is strictly convex and . Its unique minimizer is the unique solution of
The strict convexity of follows from
which can be checked by considering separately the case and the case . Since as and it has a unique minimizer which then satisfies (2.18). We further compute that
We may now solve the minimization problem (2.16).
For an arbitrary set