Lia Bronsard

• PhD
• Professor (Full) at McMaster University

In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane, is a lens cluster consisting of circular arcs containing the finite area region, attached to a single axis, w...

We study a two-dimensional inhibitory ternary system characterized by a free energy functional which combines an interface short-range interaction energy promoting micro-domain growth with a Coulomb-type long-range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a scenario in which two species are dominant...

We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a...

The Calculus of Variations is at the same time a classical subject, with long-standing open questions which have generated exciting discoveries in recent decades, and a modern subject in which new types of questions arise, driven by mathematical developments and emergent applications. It is also a subject with a very wide scope, touching on interre...

We establish, as $\rho\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $\rho$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin...

Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of $\mathbb{S}^2-$valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in $\mathbb{R}^{3}$. In part...

In this article we first prove existence of minimizers of the Landau-de Gennes energy for liquid crystals with homogeneous external magnetic field and strong uniaxial planar anchoring. Next we consider the asymptotics of solutions to the joint minimization of the energy w.r.t. the function and its boundary condition. This constitutes a generalizati...

We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a...

We study a two-dimensional quaternary inhibitory system. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which three species are vanishingly small, but interactions are correspond...

In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane, is a lens cluster consisting of circular arcs containing the finite area region, attached to a single axis, w...

We study a two-dimensional quaternary inhibitory system. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which three species are vanishingly small, but interactions are correspond...

A classical result in the study of Ginzburg-Landau equations is that, for Dirichlet or Neumann boundary conditions, if a sequence of functions has energy uniformly bounded on a logarithmic scale then we can find a subsequence whose Jacobians are convergent in suitable dual spaces and whose renormalized energy is at least the sum of absolute degrees...

Inspired by a planar partitioning problem involving multiple unbounded chambers, using classical techniques this article investigates what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. By restricting to curves which are expressible as graphs of functions, a full existence-un...

For a smooth bounded domain G⊂R3, we consider maps n:R3\G→S2 minimizing the energy E(n)=∫R3\G|∇n|2+Fs(n⌊∂G) among S2-valued map such that n(x)≈n0 as |x|→∞. This is a model for a particle G immersed in nematic liquid crystal. The surface energy Fs describes the anchoring properties of the particle and can be quite general. We prove that such minimiz...

We analyze Ginzburg--Landau minimization problems in two dimensions with either a strong or weak" tangential boundary condition. These problems are motivated by experiments in liquid crystal with boundary defects. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, whil...

We consider a sharp-interface model of $ABC$ triblock copolymers, for which the surface tension $\sigma_{ij}$ across the interface separating phase $i$ from phase $j$ may depend on the components. We study global minimizers of the associated ternary local isoperimetric problem in $\mathbb{R}^2$, and show how the geometry of minimizers changes with...

For a smooth bounded domain $G\subset\mathbb{R}^3$ we consider maps $n\colon\mathbb R^3\setminus G\to\mathbb S^2$ minimizing the energy $E(n)=\int_{\mathbb R^3\setminus G}|\nabla n|^2 +F_s(n_{\lfloor\partial G})$ among $\mathbb S^2$-valued map such that $n(x)\approx n_0$ as $|x|\to\infty$. This is a model for a particle $G$ immersed in nematic liqu...

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations...

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent \(\alpha \), under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter \(\gamma \). We show that for a wide class of density functions the energy admits a mini...

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $\alpha$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $\gamma$. We show that for a wide class of density functions the energy admits a minimizer...

We consider two nonlocal variational models arising in physical contexts. The first is the Thomas-Fermi-Dirac-von Weiz\"{a}cker (TFDW) model, introduced in the study of ionization of atoms and molecules, and the second is the liquid drop model with external potential, proposed by Gamow in the context of nuclear structure. It has been observed that...

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations...

We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau lengt...

We consider minimization problems of the Thomas–Fermi–Dirac–von Weizsäcker (TFDW) type in which the Newtonian potential is perturbed by a background potential satisfying mild conditions, which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants and obtain a more precise characterization of patte...

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. Following Choksi and Peletier, we introduce a small parameter $\eta$ to represent the...

We study a two-dimensional ternary inhibitory system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading....

We consider minimization problems of the Thomas-Fermi-Dirac-von Weizs\"{a}cker (TFDW) type, in which the Newtonian potential is perturbed by a background potential satisfying mild conditions and which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants, and obtain a more precise characterization...

We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau lengt...

We consider a variant of Gamow's liquid drop model, with a general repulsive Riesz kernel and a long-range attractive background potential with weight $Z$. The addition of the background potential acts as a regularization for the liquid drop model in that it restores the existence of minimizers for arbitrary mass. We consider the regime of small $Z...

We consider a nematic liquid crystal occupying the exterior region in R^3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external fie...

We consider a nematic liquid crystal occupying the exterior region in R^3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external fie...

We prove that both the liquid drop model in $\mathbb{R}^3$ with an attractive background nucleus and the Thomas-Fermi-Dirac-von Weizs\"{a}cker (TFDW) model attain their ground-states for all masses as long as the external potential $V(x)$ in these models is of long range, that is, it decays slower than Newtonian (e.g., $V(x) \gg |x|^{-1}$ for large...

We consider a variant of Gamow's liquid drop model, with a general repulsive Riesz kernel and a long-range attractive background potential with weight $Z$. The addition of the background potential acts as a regularization for the liquid drop model in that it restores the existence of minimizers for arbitrary mass. We consider the regime of small $Z...

We prove that both the liquid drop model in $\mathbb{R}^3$ with an attractive background nucleus and the Thomas-Fermi-Dirac-von Weizs\"{a}cker (TFDW) model attain their ground-states \emph{for all} masses as long as the external potential $V(x)$ in these models is of long range, that is, it decays slower than Newtonian (e.g., $V(x)\gg |x|^{-1}$ for...

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 B r0, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as \({|x|\to\infty}\). We...

We identify the $\Gamma$-limit of a nanoparticle-block copolymer model as the number of particles goes to infinity and as the size of the particles and the phase transition thickness of the polymer phases approach zero. The limiting energy consists of two terms: the perimeter of the interface separating the phases and a penalization term related to...

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 mino...

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau–de Gennes theory: Nematic alignment is described by a tensorial order parameter....

We analyze a non-standard isoperimetric problem in the plane associated with a metric having degenerate conformal factor at two points. Under certain assumptions on the conformal factor, we establish the existence of curves of least length under a constraint associated with enclosed Euclidean area. As a motivation for and application of this isoper...

We study vortices in p-wave superconductors in a Ginzburg-Landau setting. The state of the superconductor is described by a pair of complex wave functions, and the p-wave symmetric energy functional couples these in both the kinetic (gradient) and potential energy terms, giving rise to systems of partial differential equations which are nonlinear a...

We study the weak anchoring condition for nematic liquid crystals in the context of the Landau-De Gennes model. We restrict our attention to two dimensional samples and to nematic director fields lying in the plane, for which the Landau-De Gennes energy reduces to the Ginzburg--Landau functional, and the weak anchoring condition is realized via a p...

A thorough study of domain wall solutions in coupled Gross-Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that...

We study the structure of vortex solutions in a Ginzburg-Landau system for two complex valued order parameters. We consider the Dirichlet problem in the disk in R^2 with symmetric, degree-one boundary condition, as well as the associated degree-one entire solutions in all of R^2. Each problem has degree-one equivariant solutions with radially symme...

We consider periodic minimizers of the Lawrence-Doniach functional, which models highly anisotropic superconductors with layered structure, in the simultaneous limit as the layer thickness tends to zero and the Ginzburg-Landau parameter tends to infinity. In particular, we consider the properties of minimizers when the system is subjected to an ext...

We consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain has characteristic thickness on the scale $\eps>0$, and we consider the simultaneous limit as the thickness $\eps\rightarrow 0$ and the Ginzburg-Landau parame...

We consider the Lawrence–Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. We assume that the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes and study periodic lattice configurations in the limit as the Josephson...

We study the variational convergence of a family of two-dimensional Ginzburg- Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire...

We study minimizers of the Lawrence--Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing $s\to 0$ and the Gi...

We consider the anisotropic Ginzburg–Landau model in a three-dimensional periodic setting, in the London limit as the Ginzburg–Landau parameter k = 1/e®¥{\kappa=1/{\epsilon}\to\infty} . By means of matching upper and lower bounds on the energy of minimizers, we derive an expression for a limiting energy in the spirit of Γ-convergence. We show that,...

In this work, we study thin-film limits of the full three-dimensional Ginzburg-Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain Gamma-convergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thicknes...

We consider a Ginzburg–Landau functional for a complex vector order parameter Ψ=(ψ+,ψ−), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R2 which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one componen...

We construct local minimizers of the Gross–Pitaevskii energy, introduced to model Bose–Einstein condensates (BEC) in the Thomas–Fermi regime which are subject to a uniform rotation. Our sample domain is taken to be a solid torus of revolution in \mathbbR3{\mathbb{R}}^3 with starshaped cross-section. We show that for angular speeds ωε = O(|ln ε|)...

We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. To model experiments in which the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes, we study the structure of isolated vortices for a doubl...

We consider the three-dimensional Ginzburg–Landau model for a solid spherical superconductor in a uniform magnetic field, in the limit as the Ginzburg–Landau parameter \kappa = 1/ ɛ\rightarrow \infty . By studying a limiting functional, we identify a candidate for the lower critical field H_{c_{1}} , the value of the applied field strength at which...

We consider the two-dimensional Ginzburg-Landau model with magnetic field for a superconductor with a multiply connected cross section. We study en-ergy minimizers in the London limit as the Ginzburg-Landau parameter κ = 1/ǫ → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonze...

We consider a two-dimensional model for a rotating Bose-Einstein condensate (BEC) in an anharmonic trap. The special shape of the trapping potential, negative in a central hole and positive in an annulus, favors an annular shape for the support of the wave function u. We study the minimizers of the energy in the Thomas-Fermi limit, where a small pa...

We study a Ginzburg–Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg–Landau parameter κ = 1/ϵ→∞. The inhomogeneity is represented by a potential term V(ψ) = (a(x)−∣ψ∣2)2, with a given smooth function a(x) which is assumed to become negative in finitely many smooth subdomains, the “normally included” regions. Fo...

We study the singular limit of competition-diffusion systems in population dynamics when the initial distribution of the solution is not entirely in a domain of attraction for the system. We prove comparison principles in the viscosity sense for the solution and supersolutions to the system. By using travelling wave solutions and the distance funct...

We study a Ginzburg–Landau functional for a vector-valued order parameter that carries a spin and couples directly to the magnetic field. We verify that spin coupling reduces the lower critical field Hc1, the smallest value of the applied field strength at which vortices penetrate the superconductor, and discuss the related phenomena of a ‘spontane...

We consider a two-dimensional Ginzburg-Landau model for superconductors which exhibit ferromagnetic ordering in the superconducting phase, introduced by physicists to describe unconventional p-wave superconductors. In this model the magnetic field is directly coupled to a vector-valued order parameter in the energy functional. We show that one effe...

Recent papers in the physics literature have introduced spin-coupled (or spinor) Ginzburg–Landau models for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors and in optically confined Bose–Einstein condensates. In this Note we observe that such models can le...

We present a summary of analytical and numerical results obtained with A. J. Berlinsky and T. Giorgi on the core structure of symmetric vortices in a Ginzburg-Landau model based on S. C. Zhang’s SO(5) theory of high temperature superconductivity and antiferromagnetism. We find that the usual superconducting vortices (with normal phase in the centra...

In this paper we study the Lawrence–Doniach model for layered superconductors, for a sample with finite width subjected to a magnetic field parallel to the superconducting layers. We provide a rigorous analysis of the energy minimizers in the limit as the coupling between adjacent superconducting layers tends to zero. We identify a unique global mi...

We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. We assume that the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes and study periodic lattice configurations in the limit as the Josephson...

Symmetric vortices are finite energy solutions ψ, A to the Ginzburg–Landau equations of superconductivity with the form ψ=f(r) eidθ, A=S(r)/r2(−y, x). The existence, regularity, and asymptotic form of the solutions f(r), S(r) for any d∈Z\{0} have been established by Plohr and by Burger and Chen. In this paper we prove the uniqueness of these soluti...

We consider the problem of superconducting Ginzburg-Landau (GL) vortices with antiferromagnetic cores which arise in Zhang's SO(5) model of antiferromagnetism (AF) and high-temperature superconductivity (SC). This problem was previously considered by Arovas et al. who constructed approximate ``variational'' solutions, in the large kappa limit, to e...

We study the structure of symmetric vortices in a Ginzburg-Landau model based on S. C. Zhang's SO(5) theory of high temperature superconductivity and antiferromagnetism. We consider both a full Ginzburg-Landau theory (with Ginzburg-Landau scaling parameter kappa) and a high-kappa limiting model. In all cases we find that the usual superconducting v...

We study the asymptotic behaviour of the solution to the vector{ valued reaction{diusion equation "@t' "4' + 1 " ~ W;'(' )=0 in T ; where '" = ' : T := (0;T )! R 2 . We assume that the the potential ~ W depends only on the modulus of ' and vanishes along two concentric circles. We present a priori estimates for the solution ', and, in the spatially...

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationshi...

We describe some aspects of the Cahn-Hilliard and related equations. In particular we consider the dynamics of almost spherical interfaces and establish that almost spherical interfaces either persist forever or until they reach the boundary, a phenomenon which happens superslowly.

We study the asymptotic behavior of radially symmetric solutions of the nonlocal equation εϕ t -εΔϕ+1 εW ' (ϕ)-λ ε (t)=0 in a bounded spherically symmetric domain Ω⊂ℝ n , where λ ε (t)=1 ε∫ Ω W ' (ϕ)dx, with a Neumann boundary condition. The analysis is based on ”energy methods” combined with some a priori estimates, the latter being used to approx...

We study entire solutions on of the elliptic system where is a multiple-well potential. We seek solutions which are “heteroclinic,” in two senses: for each fixed they connect (at ) a pair of constant global minima of , and they connect a pair of distinct one dimensional stationary wave solutions when . These solutions describe the local structure o...

Let W be a potential on R2 which is equivariant by the symmetry group of the equilateral triangle and has three minima. We show that the elliptic system possesses a nontrivial smooth solution U:R2 → R2. Here DW(U)T is the transpose of the derivative DW(U).The natural energy of the problem is unbounded and compactness techniques cannot be applied. T...

A b s t r a c t . In many singularly perturbed Ginzburg–Landau type partial differ-ential equations, such as the Allen–Cahn equation, the nonlocal Allen–Cahn equa-tion, and the Cahn–Hilliard equation, the question arises whether or not the lim-iting interfaces can have high multiplicity. In other words, do there exist solutions of these PDE's with...

A finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions. The prototypical example is that of phase interfaces that meet at prescribed angles, although eutectic junctions and interactions through nonlocal effects are also considered. The method is based o...

We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and uniqueness of solutions for this model, that is, for a system of three-phase boundaries undergoing curvature motion with assigned angle conditions at the me...

We study the limiting behaviour of the solution of the Cahn-Hilliard equation using `energy-type methods'. We assume that the initial data has a `transition layer structure', i.e. uɛ≈ ± 1 except near finitely many transition points. We show that, in the limit as ɛ --> 0, the solution maintains its transition layer structure, and the transition laye...

We study the limiting behavior of the solution of with a Neumann boundary condition or an appropriate Dirichlet condition. The analysis is based on “energy methods”. We assume that the initial data has a “transition layer structure”, i.e., uϵ ≈ ±+M 1 except near finitely many transition points. We show that, in the limit as ϵ → 0, the solution main...