Schedule for: 17w5110 - Phase Transitions Models

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The transmission boundary condition which is considered appears in various exchange problems such as molecular diffusion, heat transfer between two materials, or transverse magnetization evolution in nuclear magnetic resonance (NMR) experiments. In the last context, the operator is called simplest the Bloch-Torrey equation but this is nothing else in (1D) than the complex Airy operator. We will give a detailed analysis of the spectral properties of the (various realizations of this) operator which also appears as an approximating model for the semi-classical analysis of the Schrödinger operator with a purely imaginary potential (cf the talk of Y. Almog).

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de Gennes model. The nematic is assumed to occupy the exterior of a ball, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state far from the colloid. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale, and for large balls. The relationship between the radius and the anchoring strength is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a ``Saturn ring'' defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen—Frank energy, and a dipole configuration with exactly one point defect is obtained. This is joint work with Stan Alama and Xavier Lamy.

In the recent years there has been an intensive study of a variational Landau-de Gennes (LDG) model of liquid crystals. Mathematically this appears as a higher-dimensional version of the better understood Ginzburg-Landau (GL) model of supraconductors. The main features of interest are related to the new phenomena that are specific to this type of models, for which new tools have been developed. I will present work done in the last few years together with Radu Ignat, Luc Nguyen and Valeriy Slastikov, which focuses on qualitative properties of symmetric solutions and their stability. This will be in particular an introduction to the presentations of Radu Ignat and Luc Nguyen, who will provide an in-depth, contrasting description of certain symmetry and uniqueness/multiplicity results in the GL versus LDG settings.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, we report work considering the effects of particle shape on the behavior of soft condensed matter systems [1]. We focus on orientational order, and provide a simple density functional form of the Helmholtz free energy which includes both long-range attractive and short-range repulsive interactions. We provide a detailed example, describing nematic order due to both temperature –dependent attractive (Maier-Saupe) and concentration dependent repulsive (Onsager) interactions. The shape dependence of the attractive interactions originates in the polarizability, while the shape dependence of the repulsive interactions arises through the excluded volume. We discuss the phase behavior as the relative contributions of these two effects are varied. Joint work with M. Y. Pevnyi, E.G. Virga, X. Zheng. [1] P. Palffy-Muhoray, M. Y. Pevnyi, E. G. Virga, X. Zheng. ‘The Effects of Particle Shape in Orientationally Ordered Soft Materials’, IAS/Park City Mathematics Series (to appear)

I will present a recent Gamma-convergence result that describes the behavior of the Landau-de Gennes (LdG) model for a nematic liquid crystalline film in the limit of vanishing thickness. The film is assumed to be attached to a fixed surface. In the LdG theory, an equilibrium liquid crystal configuration is specified by a tensor-valued order parameter field - a nematic Q-tensor - that minimizes an energy consisting of the bulk potential, elastic, and surface (weak anchoring) energy contributions. In the asymptotic regime of vanishing thickness, the anchoring energy plays a greater role and it is essential to understand its influence on the structure of the minimizers of the derived limiting surface energy. I will outline a general convergence result and then discuss the limiting problem in several parameter regimes. This is a joint work with Alberto Montero and Peter Sternberg.

We consider the operator ${\mathcal A}_h=-h^2\Delta+iV$ in the semi-classical limit $h\rightarrow0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.

We study a variational Ginzburg-Landau type model depending on a small parameter $\epsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $\epsilon\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $\Gamma$-limit (at the second order) as $\epsilon\to 0$. We also prove similar results for problems involving vector fields on compact surfaces embedded in $\mathbb R^3$. This is joint work with Radu Ignat.

We consider the interpretation of certain variational problems involving 1-dimensional connected sets (e.g. the Steiner-Gilbert and the Steiner tree problem) given in recent works by A. Marchese and A. Massaccesi as a Plateau problem (mass minimization) for currents with coefficients in a normed group G, a subgroup of a given Banach space E. Within this framework we discuss a natural convex relaxation procedure and variational approximations via Gamma convergence by means of functionals of phase transition / Ginzburg-Landau type. This is joint work with Mauro Bonafini (Verona) and Edouard Oudet (Grenoble).

We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phase-field model, consists of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime. The second model type is a non-linear free boundary problem for a Keller-Segel type system of PDEs in 2D with area preservation and curvature entering the boundary conditions. We find an analytic one-parameter family of radially symmetric standing wave solutions (corresponding to a resting cell) as solutions to a Liouville type equation. Using topological tools, traveling wave solutions (describing steady motion) with non-circular shape are shown to bifurcate from the standing waves at a critical value of the parameter. Our bifurcation analysis explains, how varying a single (physical) parameter allows the cell to switch from rest to motion. The work was done jointly with J. Fuhrmann, M. Potomkin, and V. Rybalko.

In this talk, I will describe recent results related to the semiclassical Robin Laplacians: Weyl asymptotics, tunneling effect and electro-magnetic Sobolev embeddings.

We minimize the standard Ginzburg-Landau functional $E_\epsilon$ over maps $u$ defined on the unit disk with values into ${\mathbb R}^3$ such that at the boundary $u$ takes values into the horizontal equator with $u=(cos k\theta, sin k\theta, 0)$ for the nonzero winding number k. It is known that the limiting problem as $\epsilon$ tends to $0$ (i.e., the harmonic map problem) has exactly two minimizers that take values either on the upper or lower hemisphere and these minimizers are symmetric. We will show that for small $\epsilon>0$, the Ginzburg-Landau functional has also exactly two minimizers that are radially symmetric. This is a joint work with Luc Nguyen, Valeriy Slastikov and Arghir Zarnescu.

We address small volume fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter $\eta$ to represent the size of the domains of the minority phase, and study the resulting droplet regime as $\eta\to 0$. A key role is played by a parameter $M$ which gives the total volume of the droplets at order $\eta^3$ and its relation to existence and non-existence of minimizers to a nonlocal isoperimetric functional on $\mathbb{R}^3$. For large values of $M$, the minority phase splits into several droplets at an intermediate scale $\eta^{1/3}$, while for small $M$ minimizers form a single droplet converging to the maximum of the confinement density.

In this talk we will consider the $p$-Laplace operator with Robin boundary conditions on Euclidean domains with sufficiently regular boundary. In particular, it will be shown how the asymptotic behavior of the first eigenvalue, when the strength of the (negative) boundary term grows to infinity, depends on the geometry of the domain. Localization of the corresponding minimizers is discussed as well. This is a joint work with Konstantin Pankrashkin.

We present the structure of a sequence $u_n: {\mathbb S}^1\to {\mathbb S}^1$ bounded in a critical space $W^{1/p,p}$, $1 (TCPL 201)

In this talk, I will review several results on the surface superconductivity in the 3-dimensional Ginzburg-Landau model. In particular, we will consider the existence of a surface energy density and study its first properties. The talk is based on joint work with Ayman Kachmar, Michael Persson Sundqvist, Xingbin Pan and Jean-Philippe Miqueu.

I will describe joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok giving criteria for existence and for non-existence of certain isoperimetric planar curves minimizing length w.r.t. a metric having conformal factor that is degenerate at two points, such that the curve encloses a specified amount of Euclidean area. These curves, appropriately parametrized, emerge as traveling waves for a bi-stable Hamiltonian system that can be viewed as a conservative model for phase transitions.

In this joint work with I.Shafrir, we show that the energy growth of a nonconstant locally minimizing solution of the Ginzburg-Landau equation in space must at least be that of the vortex filament, which is conjectured by us to be the only such solution.

One of the most striking features of superconductors without inversion symmetry is the possibility of unusual nonuniform superconducting states in the presence of a magnetic field or even without any field. Their origin can be traced to the first-order gradient terms in the Ginzburg-Landau free energy, known as the Lifshitz invariants. I will review the microscopic mechanisms leading to these terms and also discuss some of the resulting nonuniform instabilities.

The celebrated work of Onsager [1] on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions and for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it fails to provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this talk, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state in that the pressure diverges at a critical density. The energy landscape is much richer than in the low density approximation. The orientational probability distribution function in the nematic phase possesses a unique feature–it vanishes on a nonzero measure set in orientational space. Joint work with J. M. Taylor, P. Palffy-Muhoray, E. G. Virga. [1] Onsager, L., The effects of shape on the interaction of colloidal particles. Annals of the New York Academy of Sciences, 51(4), 627-659 (1949).

Applying an asymptotic method, we will establish the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains. The minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity (joint work with S. Guesmia and S. Harkat).

This talk is devoted to the semiclassical analysis of the magnetic Laplacian on a smooth domain of the plane carrying Neumann boundary conditions. We provide WKB expansions of the eigenfunctions when Neumann boundary traps the lowest eigenfunctions near the points of maximal curvature.We also explain and illustrate a conjecture of magnetic tunneling when the domain is an ellipse. This is joint work with Frédéric Hérau and Nicolas Raymond.

If a surface stabilized ferroelectric liquid crystal cell is cooled from the smectic-A to the smectic-C phase its layers thin causing V-shaped (chevron like) defects to form. These create an energy barrier that can prevent switching between equilibrium patterns. We examine a gradient flow for a Chen-Lubensky energy for which the order parameter can vanish. This allows for a transition that avoids the energy barrier. The liquid crystal can then evolve during switching in such a way that the layers melt and then heal near the chevron tip during the process. This is joint work with Lidia Mrad.

Certain spaces of Sobolev maps taking values in spheres can be decomposed into classes according to the singularities of each map. Two important examples are $W^{1,1}(\Omega;S^1)$ and $W^{1,2}(\Omega;S^2)$, where $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$. We shall present several results and open problems regarding two natural notions of distance between different classes. This is a joint work with Haim Brezis and Petru Mironescu.

Anyons are particles with quantum statistics different from those of bosons and fermions, the only known types of fundamental particles. They can occur only in low dimensions. There is sound theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons, although this remains to be unambiguously observed. 2D anyons are formally equivalent to ordinary bosons coupled to magnetic flux tubes. This leads to a many-body Hamiltonian containing long-range (and rather peculiar) two- and three-body interactions. In this talk I shall discuss a mean-field approximation of said Hamiltonian, appropriate for "almost bosonic" 2D anyons. The model we study is based on a new one-particle energy functional with self-consistent magnetic field, and the associated highly nonlinear partial differential equation. A simplified, Thomas-Fermi-like, local density model, shall be derived in a scaling limit. Joint work with Michele Correggi, Romain Duboscq and Douglas Lundholm.

I will present a recent result on the Landau-de Gennes model of liquid crystals, obtained jointly with S. Fournais and X.B. Pan. The model we studied is a functional acting on wave functions (order parameters) and vector fields (director fields) defined in a three dimensional domain. In a specific asymptotic limit of the physical parameters, we constructed critical points such that the wave function is localized near the boundary of the domain, and we computed the energy of such critical points. In physical terms, these critical points correspond to surface smectic states.

We present an overview of recent results on pseudospectra and basis properties of the eigensystem of one-dimensional Schrödinger operators with unbounded complex potentials. In particular, we address the problem of localizing the transition between spectral (Riesz basis of eigenvectors and "normal" behavior of resolvent norm) and pseudospectral (vast regions in the complex plane where resolvent norm explodes) character of these operators depending on the size of real and imaginary parts of the potential.

We study a Laudau-de Gennes model for liquid crystals where both the energy functional and the boundary data are invariant under the orthogonal group. In three dimensional settings, it is conjectured that minimizers break the rotational symmetry. We show however that in two dimensional settings, this no longer holds when the boundary data have no topological obstruction: the minimizers are `unique and rotationally symmetric'. As an application, we obtain existence of (multiple) non-minimizing rotationally symmetric critical points. Joint work with Radu Ignat, Valeriy Slastikov and Arghir Zarnescu.

In this talk, we consider Landau-de Gennes theory for liquid crystals and investigate the structure and stability of the isotropic-nematic interface in 1-D. In the absence of the anisotropic energy, we show that the uniaxial solution is the only global minimizer. In the presence of the anisotropic energy, the uniaxial solution may lose stability. We will talk about the role of the anisotropic energy term in the stability of the uniaxial solution. If time permits, we also plan to present some interesting open questions, which are related to De Giorgi conjecture. This is a joint work with Wei Wang, Pingwen Zhang and Zhifei Zhang.

We consider a non-local elliptic equation of exponential nonlinearity. The motivation for this equation is a variational problem for entropy maximization under prescribed mass and energy. We provide an unconditional existence and uniqueness proof in case of electrostatic (repulsive) self-interaction, and conditional existence and uniqueness in dimension two in the case of gravitational (attractive) self-interaction.

Nematic shells are the datum of a surface coated with a thin film of nematic liquid crystals. The interaction between the molecules and the surface induces topological defects - points where the average direction of the molecules is not well-defined. We discuss a (simplified) discrete model for nematic shells, where the molecules sit at the vertices of a triangular mesh, and study the emergence of defects, and their energetics, in the limit as the mesh parameter tend to zero. This is joint work with Antonio Segatti (Università di Pavia, Italy).

We review some recent results about the behavior of extreme type-II superconductors in the surface superconductivity regime, i.e., between the second and third critical fields. In the framework of the Ginzburg-Landau theory, we derive the effects of the (smooth) boundary curvature on the energy asymptotics and on the density of Cooper pairs. In particular, we prove the Pan's conjecture about the value of the order parameter at the boundary. We also discuss the modifications induced by the presence of corners at the boundary. Joint work with N. Rougerie, E.L. Giacomelli.

We investigate minimizers defined on a bounded domain for the Maier-Saupe energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Majumdar's modification of the Landau-de Gennes Q-tensor model, so as to constrain the competing states to take values in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we are able to use this regularity to prove that minimizers take on values strictly within the physical range.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.