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Danny Perez: So I think we can slowly reconvene and go to the next talk of the session. So our next speaker is Maria Giovanna Mora from the Universita di Pavia,
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Danny Perez: and she will tell us about the equilibrium measure for nonlocal interaction energies.
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Maria Giovanna Mora: Okay, so, can I start? Yes.
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Maria Giovanna Mora: Okay.
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Maria Giovanna Mora: Thank you very much for the introduction.
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Maria Giovanna Mora: And I would like to thank the organizers for this invitation, so what I'm going to talk about is based on a few papers
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Maria Giovanna Mora: joint with several people. So Jose Antonio Carrillo from Oxford, Joan Mateu and Joan Verdera from Barcelona,
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Maria Giovanna Mora: Luca Rondi from Milano, and Lucia Scardia from Edinburgh. So the starting point of this work—of this analysis—
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Maria Giovanna Mora: is a nonlocal interaction model motivated by dislocation theory. So, just in a few words, nonlocal interaction models are continuum models
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Maria Giovanna Mora: for larger systems of particles where each particle can interact not only with its intermediate neighbors but also with particles that are far away.
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Maria Giovanna Mora: And examples of these systems can be found in many different applications so, for instance, in biology in the study of population dynamics,
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Maria Giovanna Mora: in physics in the study of vortices in a superconductor, for instance, and in material science too. In fact, as I mentioned, our problem comes from dislocation theory. So what I'm going to do in in this talk is, well, first of all, to report on
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Maria Giovanna Mora: what we proved
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Maria Giovanna Mora: for...
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Maria Giovanna Mora: Okay, the slides are not moving. No, they are going to move. I'm still in the introduction, so
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Maria Giovanna Mora: I was saying that what I'm going to do is to
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Maria Giovanna Mora: report on the results that we have in the framework of dislocation theory, and we will see in this example that
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Maria Giovanna Mora: the anisotropy of the kernel will have a very strong impact on the structure of the minimizer,
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Maria Giovanna Mora: so on the structure of the optimal configuration of particles, and then I will [inaudible] this example to deduce some general principles on how the anisotropy of the kernel may affect the structure of the minimizer for more general
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Maria Giovanna Mora: interaction kernels. Okay, so now let me move the slide.
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Maria Giovanna Mora: Are they moving?
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Maria Giovanna Mora: Yes, okay. So, first of all, let me introduce the problem from a mathematical point of view, and then in the next slide
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Maria Giovanna Mora: we'll clarify how this is related to dislocation theory. So, the problem consists in minimizing—so in finding the minimizer—of this energy, I.
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Maria Giovanna Mora: So this is defined on probability measures in the plane, and in the application to dislocations our probability measure will represent the density—
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Maria Giovanna Mora: the density of dislocations, so the density of particles at a continuum scale.
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Maria Giovanna Mora: And so, given a certain distribution of particles, the energy of this distribution is given by this formula, so the first energy term is the typical
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Maria Giovanna Mora: nonlocal interaction energy. So, you can see that as the convolution of the kernel, W, with mu, and then you take integration with respect to mu.
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Maria Giovanna Mora: And the kernel—the interaction kernel—has a rather specific structure, so this comes from dislocation theory. I'll comment on that, in a minute.
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Maria Giovanna Mora: And so it's given by the two dimensional Coulomb kernel, so minus the logarithm, plus this anisotropy—this anisotropic term.
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Maria Giovanna Mora: And then, the second the energy term—so this one—is the so called the confinement term. So it's linear with respect to the measure, and it involves
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Maria Giovanna Mora: a potential V. So, just to have in mind, there are a couple of examples: V could be a power law like, for instance, modulus of x squared,
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Maria Giovanna Mora: or another interesting example is the one where V is the indicator function of a compact set. So I mean that V
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Maria Giovanna Mora: is zero on the compact set and plus infinity otherwise, that is, outside of the compact set. And so in this case, when you minimize this energy, I, basically you are prescribing your measures to be supported on this compact set.
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Maria Giovanna Mora: Okay, so at this stage, from a mathematical point of view, we can observe that there is an interplay between the two energy terms
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Maria Giovanna Mora: in the sense that the interaction energy has repulsive behavior because of the logarithm. So when two particles get too close to each other then
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Maria Giovanna Mora: minus the logarithm becomes very large. So the interaction energy becomes very large when these locations get too close to each other,
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Maria Giovanna Mora: while, on the other hand, the confinement—so, if you think of these two examples—the confinement becomes very large when the particles spread around the plane. So, there is
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Maria Giovanna Mora: an interplay between these two terms and the overall behavior turns out to be repulsive at short distances and attractive at larger distances, and so this is what makes
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Maria Giovanna Mora: the minimization problem interesting from a mathematical point of view.
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Maria Giovanna Mora: And, in addition, we have this anisotropy, so this, of course, does not change the singularity at zero of the kernel, which is that of the logarithm,
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Maria Giovanna Mora: but it introduces a preferred direction into the problem, and so we will see that this will have very strong impact on the structure of the minimizer.
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Maria Giovanna Mora: Okay, so as a motivation, let me mention that this energy, I, can be really seen as a continuum counterpart of discrete energy
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Maria Giovanna Mora: in the sense that you can think of dislocations as point particles, so as interacting points in the plane, and you can write down the discrete energy, the discrete level, so this is basically the discrete version of I, so we have
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Maria Giovanna Mora: the interaction and the confinement, and now, if you take the limit of these discrete energies—so the limit when the number of discrete particles tends to infinity—
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Maria Giovanna Mora: then you get exactly the energy, I. So, I is the so called the mean field limit of the discrete energies, when the number of particles goes to infinity.
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Maria Giovanna Mora: And let me also mention that the limit is in the sense of gamma convergence, so this means that minimizers of the fixed level, so minimizers of E_n,
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Maria Giovanna Mora: go to minimizers of I as n tends to infinity. So this is why we are interested in understanding minimizers of I: because they give an asymptotic description of the minimizer, the discrete minimizer, when the number of particles is sufficiently large.
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Maria Giovanna Mora: Okay, so now let me clarify how this problem is related to dislocations. So, first of all, just a few words about dislocations. So dislocations are one dimensional defects
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Maria Giovanna Mora: in the crystal structure of a metal, and they are relevant. So they are very much studied because they are considered to be
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Maria Giovanna Mora: one of the main mechanisms for plastic deformation, so for permanent deformations in metals.
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Maria Giovanna Mora: So, in reality, the geometry of dislocations may be very complicated, and so what I'm referring to here is a model of idealized dislocations
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Maria Giovanna Mora: in the sense that the I assume that all dislocation lines are straight and parallel,
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Maria Giovanna Mora: and I assume that dislocations are of edge type. So what does it mean? Well, it means that the kind of defect that we are considering is the one shown in this picture. So, here, in this picture, you can see
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Maria Giovanna Mora: a lattice, an atomic lattice,
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Maria Giovanna Mora: with some local distortion, so around here, and this distortion is due to some extra amount of atoms. So, more precisely, here there is an extra
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Maria Giovanna Mora: half plane of atoms that produces a local distortion of the lattice, so this kind of defect is called an edge dislocation.
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Maria Giovanna Mora: And the dislocation line is exactly the boundary of this extra plane of atoms, so it's the line going
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Maria Giovanna Mora: through this point and orthogonal to the screen, so it's exactly the line around which we have the distortion.
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Maria Giovanna Mora: So I'm going to assume that all defects are of this kind, with the straight and parallel dislocation lines, and actually I'm going to
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Maria Giovanna Mora: make another rather strong assumption. So I assume that all these locations have the same Burgers vector and the same sign. So this means that basically all this extra half plane of atoms are in the top part of the crystal.
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Maria Giovanna Mora: Okay, so now because of all of these geometric assumptions, we can describe
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Maria Giovanna Mora: this model in a two dimensional setting, so since all dislocation lines are parallel I can imagine, to take
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Maria Giovanna Mora: a cross section or two dimensional section of the crystal orthogonal to the dislocation lines.
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Maria Giovanna Mora: And, in this section, identify each dislocation line with its intersection with the section, so with a point, and so this is why we can
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Maria Giovanna Mora: represent dislocation lines as interacting points in the plane: because of these geometric assumptions that we that we made.
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Maria Giovanna Mora: Okay, so now we can describe the problem in a two dimensional setting, and, actually, what we do is the following: we work in a so called semi-discrete setting.
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Maria Giovanna Mora: So this means that we imagine it to be at a scale large enough so that we don't see all the details of the lattice anymore. So we don't see really the lattice anymore, we can somehow average over the lattice.
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Maria Giovanna Mora: But we still see dislocations as points, and the advantage of this semi-discrete approach is that outside of these locations we can use the continuum model that we prefer, and we can describe dislocations as
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Maria Giovanna Mora: point singularities of the strain field, and so, if you do that using
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Maria Giovanna Mora: linearized elasticity outside of dislocations and then you introduce dislocations as point singularities of the strain field,
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Maria Giovanna Mora: then you can perform this computation. You can compute, so you can compute the force that this location generates in the plane. So, imagine to have
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Maria Giovanna Mora: this location at zero. And then you can compute the force that this dislocation at zero generates—so produces—on another dislocation in the plane.
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Maria Giovanna Mora: And so, if you perform this computation in the framework of linearized elasticity, in the setting that I have mentioned, then what you find out is that this force
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Maria Giovanna Mora: is given by, well—that there is a constant, there is a material constant, kappa,
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Maria Giovanna Mora: and so, except for that the force is minus the gradient of a kernel, where the kernel is exactly the one that I've introduced before. So, this is just somehow to clarify
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Maria Giovanna Mora: the setting in dislocation theory, in which we find this specific kernel. And the fact that we have x_1 is clearly related to the fact that we are assuming all Burgers vectors to be e_1. So, this comes from this assumption. Okay, so now
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Maria Giovanna Mora: from the expression—so from the expression of this force—what can we expect from the optimal configuration of particles in this setting?
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Maria Giovanna Mora: Well, we know that if we have a dislocation located at zero and a dislocation located at x, so for instance here.
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Maria Giovanna Mora: So this dislocation at x feels a force because of the dislocation at zero given by this expression and the kernel has two components.
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Maria Giovanna Mora: One given by the logarithm and one given by the anisotropy, so the component of the force due to the gradient of the logarithm
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Maria Giovanna Mora: is represented in this picture by the red arrows and so, as you can see, this is a force—this is a repulsive force—and it has a radial direction. So the dislocation here at x, because of the dislocation at zero, is somehow pushed away
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Maria Giovanna Mora: in a radial direction, so this is exactly the repulsive behavior of the logarithm that is going to be compensated by the confinement, which is not represented in this picture.
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Maria Giovanna Mora: And then we have another component of the force due to the gradient of the anisotropy (this is represented by the blue arrows), and this is a force that is orthogonal to the radial direction, and, as you can see, points towards the vertical axis both here above the horizontal axis and below.
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Maria Giovanna Mora: So, if we have a dislocation at zero because of the anisotropic component, this is pushed on the vertical axis, either on top or on bottom of the dislocation at zero.
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Maria Giovanna Mora: So, from this picture, it is not difficult now to understand why mechanically, it is a conjecture that positive dislocations
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Maria Giovanna Mora: at equilibrium should arrange themselves in such a way to be on top of each other, so in such a way to form
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Maria Giovanna Mora: vertical alignments that are called vertical walls of dislocations. And so the first question that we try to answer is whether one can prove this conjecture rigorously from a mathematical point of view, and, so, well...
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Maria Giovanna Mora: From this picture the conjecture is is quite clear, but clearly this picture is not approved for well, first of all, because here we are looking just at a
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Maria Giovanna Mora: couple, at a pair of dislocations, while in general, we may have many dislocations and, moreover, somehow from this picture it's natural to expect to have
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Maria Giovanna Mora: somehow concentration of dislocations close the vertical axis, but here we are saying that we should really see vertical
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Maria Giovanna Mora: walls, so vertical alignment. So, going back to the energy at the continuum level, this means that minimizers—so the measures that are minimizers—should have support contained in a vertical line—so should have a one dimensional support.
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Maria Giovanna Mora: Okay, so some of this was that the starting point of our analysis.
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Maria Giovanna Mora: And well, let me first recall for you what one can prove in the case when there is no anisotropy, so the interaction is purely Coulomb. We only have the logarithmic interaction and no anisotropy, then the minimization of this energy
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Maria Giovanna Mora: is a classical problem in potential theory that goes back at least to Gauss in the framework of the electrostatics,
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Maria Giovanna Mora: and actually this energy arises in a variety of different applications such as, for instance, Coulomb gases, random matrix theory, interpolation theory and many other examples.
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Maria Giovanna Mora: And from the point of view of minimization, what you can prove is that when the confinement V blows up at infinity fast enough, then this energy
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Maria Giovanna Mora: has a unique minimizer. So the minimizer exists, is unique, is compactly supported, and then, if you assign a specific confinement V,
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Maria Giovanna Mora: in some cases, you can compute explicitly what the minimizer is. So, for instance, if V is the power law modulus of x squared,
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Maria Giovanna Mora: one can show that to the minimizer is given by the circle law, so this is the measure supported on the unit centered at zero with uniform density.
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Maria Giovanna Mora: Okay, so this is what is known when the interaction is purely Coulomb and now, well, what we would like to do is that somehow to see if these kind of results can be reproduced now, in the presence of the anisotropy.
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Maria Giovanna Mora: And so, the first thing we did was to look at the literature, so there is a huge literature about nonlocal interaction problems.
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Maria Giovanna Mora: But basically, the results that you can find in the literature can be subdivided into two groups, so the first group of results is about the explicit
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Maria Giovanna Mora: characterization of the equilibrium measure, as I mentioned in the previous slide, and you can find some of these examples
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Maria Giovanna Mora: in the book by Saff-Totik, which is a classical reference in potential theory.
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Maria Giovanna Mora: But those explicit characterization are only done for the Coulomb kernel not only in 2D, but also in higher dimension, but only for the Coulomb kernel
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Maria Giovanna Mora: and only for radially symmetric confinements. And this is because those characterizations are strongly based
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Maria Giovanna Mora: on the fact that the Coulomb kernel is the fundamental solution of the Laplacian and on the radial symmetry and clearly these are two properties that are not satisfied by our anisotopic kernel.
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Maria Giovanna Mora: And then the second group of results is about much more general interaction kernels, and so in these papers
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Maria Giovanna Mora: the goal is to establish qualitative properties of the minimizers, such as, for instance, symmetries, uniqueness, confinement and so on.
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Maria Giovanna Mora: And here there is really many, many papers. I just quoted a couple of papers that are more related to some of our settings.
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Maria Giovanna Mora: But again, a typical assumption in many of these papers is radial symmetry, and we don't have radial symmetry. And concerning
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Maria Giovanna Mora: dimensionality of minimizers—so we are thinking of the mechanical conjectures. So we would like to
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Maria Giovanna Mora: prove, for instance, that the dimension of the support of the minimizer—so our energy—is one. So, in terms of characterization of dimensionality of minimizers, the only paper we are aware of is this contribution by Balague, Carrillo, Laurent, and Raoul, where these authors
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Maria Giovanna Mora: give a bound on the dimension of the support of the minimizer in terms
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Maria Giovanna Mora: of the singularity of the Laplacian of the kernel. But unfortunately, this result does not apply to the critical singularity of the logarithm that we have,
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Maria Giovanna Mora: and moreover it's only a lower bound on the dimension; it's not a full characterization.
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Maria Giovanna Mora: In other words, we had a look at the literature and realized that because of this anisotropic term that may look rather innocuous—is just a bounded perturbation of our kernel—was not fitting into any of these frameworks.
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Maria Giovanna Mora: So, nevertheless, we were able to prove this result, and this is a joint result with Luca Rondi and Lucia Scardia.
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Maria Giovanna Mora: So, here we consider as confinement, the power law modulus of x squared. And so this is the confinement for which, in the purely Coulomb case, the minimizer is the circle law.
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Maria Giovanna Mora: And, but now we reintroduce the anisotropy in the kernel, so the kernel is exactly the one of dislocation theory.
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Maria Giovanna Mora: And so what we were able to prove is the following result, which is in the spirit of those explicit characterizations that we have seen for the Coulomb kernel.
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Maria Giovanna Mora: So, in other words we proved that for this energy, I, the minimizer exists, is unique, and it can be explicitly characterized. It's given by this measure,
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Maria Giovanna Mora: which is the semicircle on the vertical axis. So, this measure is supported on the vertical axis—more precisely on this segment: minus square root of two to square root of two, so this is a vertical segment on the vertical axis.
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Maria Giovanna Mora: And then the density is not constant but it's given by this formula. And so, if you draw a graph of this function, you get the exactly a semicircle, and so this is why the name semicircle law.
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Maria Giovanna Mora: So, actually this measure was already known in the literature, because—so if you if you really think of this measure as a measure in R—so you take its projection on the vertical axis.
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Maria Giovanna Mora: Then the semicircle law minimizes the functional where the confinement is the modulus of x squared and the kernel is minus the logarithm.
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Maria Giovanna Mora: But it is 1D. So instead of integrating with respect to R^2, you integrate on R, so it's a one dimensional problem
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Maria Giovanna Mora: with minus the logarithm as kernel and the modulus of x squared as confinement. And so what we prove the here is that this measure also minimizes this energy, which is now two dimensional, and, moreover, contains this is anisotropy.
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Maria Giovanna Mora: Okay, so let me make some other comments on this energy—on this result. So, as far as we know, this is a the first explicit characterization of our minimizer outside the very symmetric case.
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Maria Giovanna Mora: If we go back to the mechanical conjecture, while this resolved the proofs that—at least for this choice of the confinement—the minimizer has
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Maria Giovanna Mora: the vertical wall-like structure that we were expecting from mechanical considerations. And what we found very interesting is
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Maria Giovanna Mora: how this anisotropic term has dramatic effect on the shape of the minimizer. So, if there is no anisotropy—
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Maria Giovanna Mora: so we are in the purely Coulomb case—so the minimizer has two dimensional support with uniform density,
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Maria Giovanna Mora: if we introduce the anisotropy then the behavior of the minimizer is completely different. We have a one dimensional support and nonconstant density—nonuniform density.
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Maria Giovanna Mora: And,
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Maria Giovanna Mora: so we found this change of the dimensionality very interesting and so, in order to understand better how this transition—how this change of dimensionality—really occurs, we consider the following
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Maria Giovanna Mora: variant of the problem. So we keep as confinement term modulus of x squared but now we introduce a parameter alpha in front of the anisotropy
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Maria Giovanna Mora: in such a way to tune the strength of the anisotropy into the kernel. And so it's a choice to understand how the transition to the semicircle law occurs.
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Maria Giovanna Mora: And so, our goal is now to characterize the minimizers of this energy—still in the class of probability measures.
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Maria Giovanna Mora: And, of course, the questions are, well, what about existence, uniqueness, and, in particular, what about the support of minimizers. So what is the dimension of the support of the minimizer for a general alpha.
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Maria Giovanna Mora: And, actually, by symmetry arguments, it is enough to focus on the case of alpha between zero and one,
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Maria Giovanna Mora: in the sense that, once we understand what happens for those values of alpha, then, by symmetry arguments, you can then use what happens for every alpha. Maybe I'll say something about this at the end of the talk.
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Maria Giovanna Mora: So, we focus on this regime: so really somehow in the transition regime between the circle law and the semicircle law.
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Maria Giovanna Mora: Okay, so...
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Maria Giovanna Mora: so first of all... well, just a few words about existence. So actually, the presence of the parameter or, more in general, the presence of the anisotropy
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Maria Giovanna Mora: plays no role in proving the existence of minimizers. So, for existence, somehow the theory of nonlocal interaction problems is robust enough
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Maria Giovanna Mora: to now be able to deal with those kind of perturbations. And so basically here, the point is that the confinement goes to infinity at infinity fast enough.
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Maria Giovanna Mora: And so, this allows us to get the right compactness properties to establish existence of minimizers.
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Maria Giovanna Mora: And, moreover, again using this behavior of the confinement, we can also show that minimizers must be compactly supported. But this is really standard. So the presence of the anisotropy is not relevant here.
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Maria Giovanna Mora: What is a more challenging is proving uniqueness of the minimizer. So, in general, proving uniqueness may be not easy in
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Maria Giovanna Mora: this kind of problems. And so here uniqueness follows from this lemma. So this lemma says that if we take two competitors—so two probability measures
—
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Maria Giovanna Mora: compactly supported and satisfying this condition. So this is exactly what I call the interaction energy (that must be finite for both competitors). Then, if you look at the interaction on the difference of the two measures,
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Maria Giovanna Mora: this quantity has a sign, so is always nonnegative and actually is strictly positive if the two measures are different.
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Maria Giovanna Mora: So, from this lemma you can easily deduce the strict convexity of the energy, at least on the class of measures
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Maria Giovanna Mora: satisfying these two conditions: compact support and the finite interaction energy. And, since minimizers satisfy these two conditions, strict convexity on this class is enough to have uniqueness.
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Maria Giovanna Mora: So the minimizer is unique, and this is true actually also for alpha equal to one. And this is exactly the way we prove the uniqueness of the minimizer. And
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Maria Giovanna Mora: let me mention very, very briefly, how the proof of this lemma works because I'm going to refer to this at the end of the talk.
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Maria Giovanna Mora: So the idea to prove this lemma is to rewrite the interaction energy—here—in terms of the Fourier transforms in such a way that
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Maria Giovanna Mora: the convolution becomes a product and now, well, to prove that this is positive, we only have to show that the Fourier transform of the kernel is positive.
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Maria Giovanna Mora: So now, if you compute the Fourier transform of our kernels, unfortunately this is not positive, but it is positive on test functions
32:49.378 --> 32:59.908
Maria Giovanna Mora: satisfying this additional condition to be zero at zero. So on those test functions the Fourier transform has a very nice expression. So this one, now,
33:00.538 --> 33:09.718
Maria Giovanna Mora: which is a positive function for alpha in between zero and one—actually for alpha between minus one and one because we have these two coefficients.
33:10.618 --> 33:16.888
Maria Giovanna Mora: And now positivity on this class of test functions is enough because
33:17.458 --> 33:27.808
Maria Giovanna Mora: in the lemma we are looking at measures that are a difference of two probability measures. So, this measure nu has an integral equal to zero.
33:28.318 --> 33:45.418
Maria Giovanna Mora: And so, this means that its Fourier transform is zero at zero. And so the test function that we have here is a positive test function that is zero at zero, and so we just need to have positivity on this class of test functions. So this is enough
33:46.348 --> 33:56.458
Maria Giovanna Mora: to get the lemma. And here, let me note that alpha equal to 1 and, symmetrically, alpha equal to -1,
33:57.748 --> 34:10.678
Maria Giovanna Mora: exactly those values for which—here in the Fourier transform—we have somehow a loss of coercivity. So if alpha is equal to 1 or alpha is equal to -1
34:11.068 --> 34:18.808
Maria Giovanna Mora: somehow we got some degeneracy of the Fourier transform of the kernel, and so I'll go back to these at the end of the talk.
34:19.918 --> 34:35.968
Maria Giovanna Mora: Okay, so, so far what we know for alpha in between 0 and 1 is that the minimizer that exists is unique and it is compactly supported. What else can we say about the minimizer? Well, we know that the energy is convex
34:37.108 --> 34:45.478
Maria Giovanna Mora: and so, since we have convexity, minimality is equivalent to the Euler-Lagrange equations—so the first order
34:45.808 --> 34:55.648
Maria Giovanna Mora: necessary conditions for minimality—they are also sufficient conditions for minimality because we have convexity. And in our setting
34:56.248 --> 35:03.838
Maria Giovanna Mora: the Euler-Lagrange questions take this form. So let us call mu_alpha the minimizer.
35:04.828 --> 35:14.068
Maria Giovanna Mora: Then what we have is that this function—so this is called the potential function—so it's the convolution of the kernel with the minimizer.
35:14.788 --> 35:24.448
Maria Giovanna Mora: So the potential function plus 1/2 of the confinement—this function has to be constant on the support of the minimizer.
35:25.198 --> 35:44.248
Maria Giovanna Mora: And it has to be above this constant elsewhere in the plane. So these two conditions together are equivalent to minimality. And so to find the minimizer, we have to find the unique measure mu_alpha satisfying
35:45.478 --> 35:47.398
Maria Giovanna Mora: these two conditions.
35:48.448 --> 36:01.978
Maria Giovanna Mora: And so now the question is, can we identify such a measure? So when we first now faced this problem, we tried to make some guesses on what the minimizer
36:03.208 --> 36:09.208
Maria Giovanna Mora: could be. And, well, somehow the first—somehow the easiest guess that you can do
36:09.598 --> 36:19.588
Maria Giovanna Mora: is that the support of the minimizer is the domain enclosed by an ellipse. So, because, well, we know that for alpha equal to zero—so no anistropy—
36:20.008 --> 36:27.958
Maria Giovanna Mora: the minimizer is supported on a unit ball, and then for alpha equal to one, we know that we get
36:28.708 --> 36:38.758
Maria Giovanna Mora: a measure supported on the vertical axis. So for alpha in between 0 and 1, maybe we should get something in between. So maybe an ellipse with major axis on
36:39.538 --> 36:47.758
Maria Giovanna Mora: the vertical axis. So somehow this is the easiest test. Also because, if you remember, the picture about the force
36:48.238 --> 37:03.058
Maria Giovanna Mora: generated by a dislocation, so the force due to the anisotropy has this component pointing towards the vertical axis. So, somehow it's reasonable to expect a concentration close to the vertical axis.
37:04.168 --> 37:15.478
Maria Giovanna Mora: Okay, but even if the support of the measure is a two dimensional support, what can we say about the density? Can we expect the density
37:15.868 --> 37:26.458
Maria Giovanna Mora: of the minimizer to be constant? So, in other words, can we expect the minimizer to be the normalized characteristic function of a three dimensional domain?
37:26.968 --> 37:32.038
Maria Giovanna Mora: Well, let's have a look at what we know from the Euler-Lagrange equations. So assume
37:32.848 --> 37:43.078
Maria Giovanna Mora: the measure—the minimizer—to be the characteristic function of a set. So, for instance, of a two dimensional ellipse normalized by its area.
37:43.738 --> 37:53.818
Maria Giovanna Mora: And then let's try the kernel as the logarithm plus alpha times the anisotropy. Okay, so the first Euler-Lagrange equation says that
37:54.358 --> 38:04.678
Maria Giovanna Mora: this function—so this is the potential—plus—this is the confinement divided by two—has to be constant inside the support. So, in particular, inside the ellipse.
38:05.848 --> 38:25.228
Maria Giovanna Mora: So, let's take the Laplacian of both sides. Well, W_0 is the Coulomb kernel—it's the fundamental solution of the Laplacian. So when we take the Laplacian of this quantity, we get a constant times the measure itself
38:26.488 --> 38:40.978
Maria Giovanna Mora: because W_0 is the fundamental solution of the Laplacian. Then we get alpha times the Laplacian of the anisotropic part. This quantity has a Laplacian equal to two. And then the Laplacian of the constant is equal to zero.
38:42.268 --> 38:56.128
Maria Giovanna Mora: But now, inside the ellipse, we are assuming the measure to have uniform density, so this new alpha is a constant. So, this is constant. This is constant. So, this means that
38:56.518 --> 39:04.168
Maria Giovanna Mora: this Laplacian has to be constant. So the Laplacian of the anisotropic kernel—of the anisotropic part of the kernel—
39:04.648 --> 39:10.318
Maria Giovanna Mora: convolution the characteristic function of the ellipse has to be constant inside the ellipse.
39:11.008 --> 39:29.278
Maria Giovanna Mora: And so, well, when we saw this condition we that that this was really weird. Why should this Laplacian be constant inside the ellipse? And so our first computations were somehow motivated by disproving that minimizers
39:31.108 --> 39:36.928
Maria Giovanna Mora: can be characteristic functions of ellipses. So we wanted—we tried to show that this Laplacian
39:38.128 --> 39:45.148
Maria Giovanna Mora: cannot be—is, in general, not constant. But, in doing so, we actually realized that
39:46.018 --> 39:57.568
Maria Giovanna Mora: this Laplacian is actually constant and this is true for any ellipse. So if mu_alpha is the characteristic function of an ellipse, this Laplacian is constant inside the ellipse.
39:58.348 --> 40:17.368
Maria Giovanna Mora: And in doing so we realized that, in fact, that the minimizer is the characteristic function of an ellipse. And so, more precisely, this is the main results that we proved. So let alpha in between zero and one, and let us consider this set. So this is
40:18.508 --> 40:27.808
Maria Giovanna Mora: the ellipse with center at the origin and semi-axis square root one minus alpha, square root one plus alpha.
40:28.318 --> 40:39.328
Maria Giovanna Mora: So, let's take the measure given by the characteristic function of this two dimensional ellipse normalized by area because we are working with probability measures.
40:39.988 --> 40:49.348
Maria Giovanna Mora: Then, this measure mu_alpha satisfies the two Euler-Lagrange equations. And so it's the unique minimizer of the energy.
40:50.218 --> 41:01.858
Maria Giovanna Mora: So the global picture is the following. For alpha equal to zero—so purely Coulomb interaction—the minimizer is given by uniform distribution on the unit ball.
41:02.458 --> 41:14.458
Maria Giovanna Mora: Now, if you introduce a little bit of anisotropy, you still get a uniform distribution but on a two dimensional ellipse with the major axis on the vertical axis.
41:15.118 --> 41:24.748
Maria Giovanna Mora: Then, as alpha increases and approaches one, the distribution stays uniform, but the ellipse gets more and more elongated vertically.
41:25.108 --> 41:37.078
Maria Giovanna Mora: And it's only at alpha equal to one that we see a drop of dimensionality and we get the semicircle law on the vertical axis—so with one dimensional support.
41:37.828 --> 41:44.548
Maria Giovanna Mora: And the situation is completely—so the picture is completely symmetrical for alpha in between minus one and zero.
41:44.998 --> 42:03.658
Maria Giovanna Mora: So for alpha strictly between minus one and zero, we get the uniform distribution on a two dimensional ellipse with major axis on the horizontal axis. And then at alpha equal to minus one, we get a drop of dimension and we get the semicircle law on the horizontal axis.
42:05.128 --> 42:19.468
Maria Giovanna Mora: And so, this is the global picture. And let me mention a recent result that shows somehow that in this context ellipses are really generic
42:20.278 --> 42:34.318
Maria Giovanna Mora: in the sense that while keeping the same confinement term, modulus of x squared, we can prove the following. So, instead of the anisotropy
42:34.948 --> 42:54.388
Maria Giovanna Mora: coming from dislocation theory, we can actually consider a much more general kernel, but it must be even and zero homogeneous. So this is somehow the two crucial properties that the dislocational anisotropy has. So, if we
42:55.408 --> 43:01.618
Maria Giovanna Mora: perturb the logarithm by a kernel with these two properties: even, zero homogeneous, and it's smooth enough,
43:02.398 --> 43:13.738
Maria Giovanna Mora: then we can show that the corresponding energy has a unique minimizer given by the normalized characteristic function of a two dimensional ellipse.
43:14.218 --> 43:22.708
Maria Giovanna Mora: This is true if the perturbation—if the kernel K—is small enough in a suitable sense.
43:23.548 --> 43:44.668
Maria Giovanna Mora: So for the anisotropy given by dislocation theory, we have a global picture. For these more general kernels, we can guarantee that the minimizer is an ellipse, and we can do that only for small perturbation. So, this is true only for K controlled in a suitable norm.
43:45.988 --> 43:58.198
Maria Giovanna Mora: Okay, so now let me get to the end of the talk, and let me try to draw some conclusions on what we have seen. Well...
43:59.698 --> 44:12.178
Maria Giovanna Mora: What we found out is that while alpha equal one and symmetrically alpha equal to minus one correspond to a drastic change of dimensionality.
44:12.748 --> 44:23.188
Maria Giovanna Mora: And, of course, it would be a very interesting to understand the reason why this happens. And while we saw before that these two values
44:23.728 --> 44:34.408
Maria Giovanna Mora: are critical values of the Fourier transform—those are values at which you have some degeneracy of the Fourier transform—or correspondingly
44:35.038 --> 44:44.428
Maria Giovanna Mora: the degeneracy of the differential operator associated with the kernel W_alpha, in the sense that here—on the right hand side—
44:44.878 --> 44:56.458
Maria Giovanna Mora: we have a sort of anisotropic Laplacian of the delta, and for alpha equal to one or minus one this operator on the right hand side becomes degenerate.
44:57.598 --> 45:12.328
Maria Giovanna Mora: So, a natural question is now the following, "well, is there a relationship between these two facts: can we deduce a general principle relating the degeneracy of the Fourier transform to the loss of dimensionality?"
45:13.348 --> 45:26.548
Maria Giovanna Mora: And so we want to get some insights into this problem. We tried to reproduce this behavior we have seen in higher dimensions, so in R^n.
45:27.628 --> 45:42.868
Maria Giovanna Mora: And, well, a natural choice in R^n for our potential is the following: so we replace the logarithm by the Coulomb kernel in R^n, which is this one up to constants.
45:43.618 --> 45:57.658
Maria Giovanna Mora: And then, a natural choice of the anisotropy is this one. Why is this? Well because, for instance, if you look at the associated the differential operator,
45:58.138 --> 46:15.148
Maria Giovanna Mora: you see a completely analogous behavior to the one we had in 2-d. And now the critical values of the parameter are (n-2) and (-n+2), and the structure is completely analogous.
46:16.318 --> 46:28.378
Maria Giovanna Mora: And, in fact, for this kernel, which is now—so for this problem, which is now in n dimensions—so the confinement is still modulus of x squared, but
46:28.828 --> 46:41.458
Maria Giovanna Mora: now, the problem is in R^n and this is the interaction kernel. So what we were able to prove is the following. So, let us look for simplicity at the cases where the parameter alpha is positive.
46:42.268 --> 46:52.678
Maria Giovanna Mora: Then, for alpha in between zero and the critical value, n-2. The minimizer—so we prove that the minimizer is unique and
46:53.098 --> 47:00.808
Maria Giovanna Mora: is given by an n-dimensional ellipsoid. So, it's the domain enclosed by an n-dimensional ellipsoid.
47:01.258 --> 47:16.768
Maria Giovanna Mora: But this is true, up to the critical value, n-2, included. So this means that at the critical value, we don't have loss of dimensionality, but the minimizer is still fully n-dimensional.
47:18.358 --> 47:26.908
Maria Giovanna Mora: So, in other words, there is no drop of dimensionality in this is generalization of this
47:27.778 --> 47:38.578
Maria Giovanna Mora: extension to any dimension. And so, well, we can conclude that, at least for smooths confinements, like modulus of x squared,
47:39.058 --> 47:49.348
Maria Giovanna Mora: the degeneracy of the Fourier transform does not imply loss of dimensionality. What is probably true—and we have some partial results on this—
47:49.948 --> 48:09.088
Maria Giovanna Mora: is that as long as you don't have degeneracy—so if you don't have degeneracy, so if your Fourier transformer is non-degenerate—then you shouldn't see a loss of dimensionality. So as long as you don't have degeneracy you should get a minimizer, which is full dimensional.
48:10.438 --> 48:26.308
Maria Giovanna Mora: Okay, this is for everything I said that was about the confinement modulus of x squared. What happens for other confinements? So, for instance, what happens if we
48:27.688 --> 48:53.818
Maria Giovanna Mora: get rid—so we drop the modulus of x squared term and we replace it by a physical confinement. So, let's get back to 2D and assume now that the support of the measure must be contained, for instance, in an ellipse. Then what we were able to prove is that—
48:54.838 --> 49:03.688
Maria Giovanna Mora: So here I'm referring to the dislocation anisotropy. What we were able to prove is that, in this case, the minimizer exists, is unique,
49:04.258 --> 49:15.088
Maria Giovanna Mora: and it's supported on the boundary of the ellipse. But what is more surprising is that, in this case, the minimizer is the same for every alpha.
49:15.778 --> 49:28.798
Maria Giovanna Mora: So, in other words the minimizer is completely insensitive to the anisotropy, and so the minimizer that we get for the purely Coulomb interaction—so for alpha equal to zero, which is, by the way
49:29.518 --> 49:46.678
Maria Giovanna Mora: an explicit measure—so a known measure—is the minimizer also for every alpha between minus one and one. And so, somehow this result is telling us that for other choices of... Well, first of all
49:47.698 --> 49:57.298
Maria Giovanna Mora: ellipses play a very relevant role in this in this problem, so they are really generic, and
49:58.918 --> 50:10.438
Maria Giovanna Mora: on the other hand, this is telling us that, if we change confinements, then the behavior of the minimizer also in the presence of anisotropy can be completely different. So while
50:11.128 --> 50:21.298
Maria Giovanna Mora: for modulus of x squared, we saw a change of dimensionality of the minimizer, and, in any case, we saw that the anisotropy was really
50:21.808 --> 50:29.638
Maria Giovanna Mora: changing the shape of the minimizer. For this physical confinement, the minimizer is completely insensitive to
50:30.238 --> 50:42.028
Maria Giovanna Mora: the anisotropy, so the behavior is completely different. Okay, so I thank you for your attention and I will be happy to answer your questions, if you have.
50:43.558 --> 50:44.008
Maria Giovanna Mora: Thank you.
50:46.378 --> 50:52.168
Danny Perez: Thank you so much, Maria Giovanna, for this very interesting talk. Do we have some questions?
50:58.228 --> 51:00.268
Raghav Venkatraman: So, really beautiful talk.
51:01.468 --> 51:03.538
Raghav Venkatraman: Can—I have a couple of questions.
51:05.368 --> 51:06.478
Raghav Venkatraman: So one is
51:07.588 --> 51:21.178
Raghav Venkatraman: in the Sandier-Serfaty type of thing without the anisotropy, of course, they look at sort of the next order and they can show that somehow minimizers have this crystallization type of behavior
51:22.228 --> 51:28.318
Raghav Venkatraman: that you observe at the next order. So, can you—I mean—can one sort expect something like that or
51:31.108 --> 51:37.678
Raghav Venkatraman: that you don't have any kind of crystallization or some chain formation or something like—can you say something about that?
51:38.188 --> 51:46.468
Maria Giovanna Mora: Well, this is a very interesting question, in fact, this is something that we would like to do. So, the idea would be—so somehow...
51:48.028 --> 51:53.758
Maria Giovanna Mora: we know now the minimizer [inaudible] of the limit, and so this gives
51:55.528 --> 52:00.178
Maria Giovanna Mora: an indication of the overall behavior of particles
52:01.798 --> 52:08.128
Maria Giovanna Mora: when the number is very big, but it will be interesting to understand how on the discrete level
52:08.638 --> 52:19.288
Maria Giovanna Mora: the semicircle is reached—so is approximated. And so, one way to do it would be exactly to do something in the spirit of Sandier-Serfaty's
52:19.948 --> 52:30.628
Maria Giovanna Mora: papers, and so look at some higher order terms in some sort of gamma expansion. So this is something that we would like to do but
52:31.678 --> 52:52.798
Maria Giovanna Mora: it's much more difficult than in the Sandier-Serfaty case. Somehow the problem which is behind is vector valued. And because it involves somehow the equilibrium equations for linearized elasticity, and it's also invariant by
52:53.878 --> 53:05.488
Maria Giovanna Mora: skew-symmetric matrices, and so also coordinate [inaudible] will play a role. We thought about it and realized that it's very difficult. We would like to do it, but we still don't have a
53:06.238 --> 53:19.558
Maria Giovanna Mora: complete result, and this is certainly something that we would like to do. And, well, I didn't show in this talk any numerical tests. We have some simulations where you can see that, for instance, in—
53:20.788 --> 53:32.788
Maria Giovanna Mora: for alpha between zero and one, where you get those ellipses, from the numerical tests, it seems that we have some sort of vertical alignment—some patterns. So it seems that we see some patterns.
53:33.328 --> 53:46.228
Maria Giovanna Mora: So it would be interesting to see if this is just an effect of our simulations that maybe are not particularly good or there is really something behind behind it. So yeah, so this is somehow
53:47.248 --> 53:49.258
Maria Giovanna Mora: the state of the art.
53:51.328 --> 53:54.208
Raghav Venkatraman: Another sort of, I guess, related question is, I mean,
53:55.678 --> 54:01.978
Raghav Venkatraman: the Coulomb kernel, of course, is somehow very isotopic is it conceivable to
54:03.028 --> 54:11.458
Raghav Venkatraman: look at the Green's function of some other sort of anisotropic elliptic operator, so minus divergence of A grad U.
54:12.898 --> 54:23.068
Raghav Venkatraman: Like if A is even just constant but different constants along the diagonal, I mean, you might conceivably get how ellipses might play some sort of a role. Do you think that
54:23.818 --> 54:35.368
Raghav Venkatraman: is some something that might have something to—there might be some underlying effect of a differential operator like that which takes takes into account some anisotropy like that?
54:36.058 --> 54:37.618
Maria Giovanna Mora: Well, yeah I think that,
54:39.148 --> 54:51.598
Maria Giovanna Mora: so, for instance, for the things you are mentioning—so just multiplying the gradient by a constant—I think that some of that should be feasible. So I think that some of these results
54:53.158 --> 54:53.908
Maria Giovanna Mora: can be
54:55.228 --> 54:56.518
Maria Giovanna Mora: still proved.
54:57.538 --> 55:06.448
Maria Giovanna Mora: Of course, well, concerning the explicit characterizations of the minimizer—well, probably, yes, something can be done, but
55:06.928 --> 55:14.908
Maria Giovanna Mora: of course, our result is a bit rigid, in the sense that we really—so except for the perturbation result that I mentioned at the end—the
55:15.178 --> 55:25.948
Maria Giovanna Mora: explicit characterization is really based on explicit computations. And so, if you can change a little bit things, but if you change that much then
55:27.538 --> 55:35.608
Maria Giovanna Mora: I'm not sure what happens. So, something in the direction you mentioned can be done.
55:37.228 --> 55:37.828
Maria Giovanna Mora: But,
55:38.668 --> 55:41.758
Maria Giovanna Mora: that is what I know. Thank you for your questions.
55:42.658 --> 55:43.078
Danny Perez: Thank you.
55:44.308 --> 55:45.538
Danny Perez: You have another question from Irene.
55:49.168 --> 55:50.038
Irene Fonseca: Hi, Maria Giovanna.
55:52.378 --> 55:56.548
Irene Fonseca: So I have actually a couple of questions on your last slide.
56:00.538 --> 56:19.468
Irene Fonseca: For the confinement, should you go from confinement of polynomial type to one that has good support—it's very different, right? But what if you change your confinement to still maybe something that looks like polynomial growth at infinity? Would it be different? That is one question. And the second one is
56:20.908 --> 56:42.448
Irene Fonseca: so when you generalize your anisotropy for the plane, you said well, okay, I can take a function, which is homogeneous of degree zero and it has some control of derivatives. But here in dimension n, you lost that homogeneity, right? It's x_1 squared over |x|^n.
56:44.608 --> 56:45.418
Maria Giovanna Mora: Okay, so,
56:46.468 --> 56:58.768
Maria Giovanna Mora: yeah. For the first question, so, for instance, one could try to to consider as confinement another power law, like modulus of x to the power p.
56:59.848 --> 57:18.118
Maria Giovanna Mora: So, for the logarithm the minimizer is known. In this case, too. And it's still supported on a ball and the density is not constant. And so you get something radial and
57:19.828 --> 57:27.598
Maria Giovanna Mora: it's not constant and the fact that it is not constant—well, you can see it from the Euler-Lagrange equations.
57:29.128 --> 57:29.698
Maria Giovanna Mora: Here.
57:30.808 --> 57:32.008
Maria Giovanna Mora: So if you...
57:33.358 --> 57:45.988
Maria Giovanna Mora: If you write your Euler-Lagrange equations and now you take the Laplacian, the logarithm will give you something—a constant. If you get
57:46.768 --> 57:55.468
Maria Giovanna Mora: the characteristic function of an ellipse—here—you would get something constant. And this is the property of our anisotropy.
57:55.738 --> 58:05.968
Maria Giovanna Mora: But now, if you change the power law—here—you take the Laplacian—here—you don't get a constant. So this is telling you that if you change the power law
58:07.738 --> 58:23.398
Maria Giovanna Mora: and you keep the anisotropy then the support of the minimizer could be an ellipse but the density cannot be uniform, and so it becomes much more difficult to guess [what] the minimizer
58:24.448 --> 58:46.018
Maria Giovanna Mora: could be. So we did some numerical tests, so it seems that the support is still an ellipse but with nonuniform density. But then it's very difficult to understand what it could be. And concerning your second question, somehow what is relevant in this perturbation argument
58:47.518 --> 59:02.758
Maria Giovanna Mora: is that when you take the Laplacian of K—or let's say second derivatives of K—you get a kernel which is minus two homogeneous.
59:03.598 --> 59:15.388
Maria Giovanna Mora: And two is exactly the dimension of the space. And what you can show is that if you have an even kernel which is minus two homogeneous,
59:16.258 --> 59:23.398
Maria Giovanna Mora: and you take the convolution with a characteristic function of an ellipse, you get something which is constant inside the ellipse. So somehow
59:24.088 --> 59:49.318
Maria Giovanna Mora: the relevant properties are those for second derivatives. And so in 2D we need even and minus two homogeneous, and two is the dimension of the space we are working in. And now, when you go to dimension n, you have exactly the same picture. So this is now
59:51.028 --> 01:00:07.078
Maria Giovanna Mora: (2-n) homogeneous, and when you take two derivatives you get minus n homogeneous, where n is the dimension of the space. So this structure is the same, and, somehow, this is what is relevant for this result.
01:00:07.678 --> 01:00:08.968
Irene Fonseca: Okay. Got it, thanks.
01:00:10.798 --> 01:00:13.948
M.CARME CALDERER: Yeah, I have a question that I wrote in the chat but
01:00:14.038 --> 01:00:35.098
M.CARME CALDERER: you didn't see it. So could you generalize your method to include maybe some, say, small in some way attractive contribution in W and maybe an attractive contribution that acts on a smaller length scale than the repulsive part.
01:00:35.608 --> 01:00:37.078
M.CARME CALDERER: The reason is that
01:00:37.288 --> 01:00:43.108
M.CARME CALDERER: this would be—I think—quite interesting for some systems where you need,
01:00:43.948 --> 01:01:03.478
M.CARME CALDERER: say, other kinds of laws than beyond the Coulomb repulsion, for example. And in biology this could be a very interesting issue to look at. So if you have some kind of idea of how we could go about maybe extending it or generalizing it, perhaps as motivation say.
01:01:04.888 --> 01:01:05.248
Maria Giovanna Mora: Well...
01:01:06.298 --> 01:01:08.998
Maria Giovanna Mora: It really depends on what you
01:01:10.228 --> 01:01:21.838
Maria Giovanna Mora: want to prove in the sense that, well, some of these results, like so existence, are very robust and so probably they can they can be carried through
01:01:22.078 --> 01:01:27.988
Maria Giovanna Mora: without problems, but for the explicit characterization, the structure is very important.
01:01:28.378 --> 01:01:32.068
Maria Giovanna Mora: So it would really depend on
01:01:33.568 --> 01:01:43.978
Maria Giovanna Mora: the specific structure of the attractive terms that you have in mind because, as I said before, those explicit characterization are very rigid. So also
01:01:45.058 --> 01:02:01.438
Maria Giovanna Mora: related to Irene's question, if you change the confinement, then everything changes. So it really depends on the structure that you have in mind, and that would be definitely something interesting to look at.
01:02:02.998 --> 01:02:09.598
M.CARME CALDERER: So, for instance Lennard-Jones kind of repulsion—that would also have a small attraction at
01:02:10.798 --> 01:02:11.758
M.CARME CALDERER: a positive scale.
01:02:12.418 --> 01:02:12.928
Maria Giovanna Mora: Okay.
01:02:13.228 --> 01:02:16.258
M.CARME CALDERER: It would be really very interesting in terms of applications.
01:02:17.098 --> 01:02:20.458
Maria Giovanna Mora: Okay, I see. Thank you. Well, I don't know.
01:02:22.438 --> 01:02:22.858
Maria Giovanna Mora: Thanks.
01:02:23.998 --> 01:02:25.768
Danny Perez: Alright, we'll take one last question.
01:02:26.098 --> 01:02:34.018
Maciej Buze: Hi, hello, very nice. Thank you. So I'm a former PhD student of Christoph Ortner and I'm currently working on the
01:02:34.618 --> 01:02:50.908
Maciej Buze: very related problem of near-crack-tip plasticity. So on your length-scale that would be just like your problem, except that instead of it being posed on R^2, you would have R^2 except for the crack, which is a line.
01:02:51.298 --> 01:02:53.098
Maciej Buze: So basically
01:02:55.078 --> 01:02:59.938
Maciej Buze: my question is so on your last slide, you said you have this physical containment. So
01:03:01.558 --> 01:03:24.058
Maciej Buze: for the crack problem, you also look at the Green's function, and then you have an explicit Green's function, and then it is a function in full two variables. So that would translate to your W not being a function of the difference of two variables, but the function fully in two variables.
01:03:24.538 --> 01:03:25.228
Maria Giovanna Mora: I see, okay.
01:03:26.668 --> 01:03:34.798
Maciej Buze: I suppose, when you say physical confinement, I guess, if you look at the the domain that was, let's say, a ball, for instance,
01:03:35.248 --> 01:03:41.158
Maciej Buze: and you just looked at the Green's function, which is also explicitly given. That would give you sort of
01:03:41.608 --> 01:03:58.408
Maciej Buze: physical confinement, I suppose, because it would be a truly geometrical constraint. Is that what you mean when you say "physical confinement" or do you simply mean that you have a, as you said in the early parts of your talk, where you look at the indicator function of...
01:03:58.858 --> 01:04:07.018
Maria Giovanna Mora: Yeah, so actually this is what I meant. So that's somehow thinking of the application to dislocations.
01:04:07.588 --> 01:04:12.088
Maria Giovanna Mora: So the particles really are physically confined
01:04:12.298 --> 01:04:13.588
Maria Giovanna Mora: into a set.
01:04:14.068 --> 01:04:16.498
Maria Giovanna Mora: So this is what I actually mean
01:04:18.598 --> 01:04:21.718
Maria Giovanna Mora: with this terminology, so they are really physically confined.
01:04:22.888 --> 01:04:32.368
Maciej Buze: But my point is, so in near-crack-tip plasticity I also look at dislocations interacting near the crack-tip. And the point is that, if you look at the
01:04:33.388 --> 01:04:45.268
Maciej Buze: Green's function corresponding to your geometry, which in your case I guess is just some set finite set. Then my point, basically, is that you can derive
01:04:46.828 --> 01:04:53.038
Maciej Buze: justified physical confinement, which comes from the geometry, as opposed to just—
01:04:54.058 --> 01:05:02.098
Maciej Buze: So, have you thought of that? Is that something? Is it? Because I suppose as soon as you go into a setup where your W is
01:05:03.238 --> 01:05:13.558
Maciej Buze: you know, fully a function in two variables, then everything gets very complicated, and it could well be that it's just beyond the reach of current methods. Is that a reasonable comment?
01:05:14.968 --> 01:05:19.048
Maria Giovanna Mora: Well, that's for sure. Actually I don't have a
01:05:20.818 --> 01:05:21.838
Maria Giovanna Mora: reasonable answer unfortunately,
01:05:24.268 --> 01:05:25.348
Maria Giovanna Mora: in the sense that
01:05:26.638 --> 01:05:28.078
Maria Giovanna Mora: I don't know. So,
01:05:29.398 --> 01:05:32.008
Maria Giovanna Mora: I should think about it. So I...
01:05:33.568 --> 01:05:37.618
Maciej Buze: I'm in touch with Lucia Scardia as well, so perhaps we can think about this together.
01:05:38.998 --> 01:05:39.628
Maciej Buze: Very nice talk.
01:05:40.228 --> 01:05:46.318
Maria Giovanna Mora: We can get in touch through her, and maybe I can have a better look on what you are doing and what you are interested in. Thank you very much.
01:05:52.738 --> 01:05:57.868
Danny Perez: Right, well, that was a very nice discussion. So thanks to Maria Giovanna and thanks, Tony, for a very nice morning.