arXiv subjects

Stan Alama

Publications and source records attributed to Stan Alama.

At least 19 recordsLinked to original sources

arXiv2023

The standard lens cluster in R^2 uniquely minimizes relative perimeter

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, with two triple junctions where the arcs meet at 120 degree angles. In particular, we show that such a configuration is a local minimizer of the total perimeter functional, and on the other hand any local minimizer of perimeter among clusters with the given area constraints must coincide with a lens cluster having this geometry. Some known results and conjectures on similar problems with both finite and infinite area constraints are presented at the conclusion.

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arXiv2023

$\Gamma$-Convergence of the Ginzburg-Landau Functional with tangential boundary conditions

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 of vortices. However, the corresponding question for the case of tangential or normal boundary conditions has not been considered. In addition, the question of convergence of up to the boundary is not very well understood. Here, we consider these questions for a bounded, connected, open set of $\mathbb{R}^{2}$ with $C^{2,1}$ boundary.

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arXiv2023

On minimizers of the 2D Ginzburg-Landau energy with tangential anchoring

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, while both boundary and interior vortices are possible for strong anchoring in the first order limit.

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arXiv2022

Far-Field Expansions for Harmonic Maps and the Electrostatics Analogy in Nematic Suspensions

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 liquid crystal. The surface energy $F_s$ describes the anchoring properties of the particle, and can be quite general. We prove that such minimizing map $n$ has an asymptotic expansion in powers of $1/r$. Further, we show that the leading order $1/r$ term is uniquely determined by the far-field condition $n_0$ for almost all $n_0\in\mathbb S^2$, by relating it to the gradient of the minimal energy with respect to $n_0$. We derive various consequences of this relation in physically motivated situations: when the orientation of the particle $G$ is stable relative to a prescribed far-field alignment $n_0$; and when the particle $G$ has some rotational symmetries. In particular, these corollaries justify some approximations that can be found in the physics literature to describe nematic suspensions via a so-called electrostatics analogy.

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arXiv2020

A nonlocal isoperimetric problem with density perimeter

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 $α$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $γ$. We show that for a wide class of density functions the energy admits a minimizer for any value of $γ$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $γ$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $γ\to 0$ limit corresponds, under a suitable rescaling, to a small mass $m=|Ω|\to 0$ limit when $p d-α+1$.

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arXiv2020

Convergence of the TFDW Energy to the Liquid Drop Model

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 the two models exhibit many of the same properties, especially in regard to the existence and nonexistence of minimizers. We show that, under a "sharp interface" scaling of the coefficients and constrained mass, the TFDW energy Gamma-converges to the Liquid Drop model, for a general class of external potentials. Finally, we present some consequences for global minimization of each model.

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arXiv2020

Saturn ring defect around a spherical particle immersed in nematic liquid crystal

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 of the Landau-de Gennes $Q$-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.

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arXiv2020

Thin Film Liquid Crystals with Oblique Anchoring and Boojums

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 length scale parameter,) defects will occur in the interior, as in the case of strong (Dirichlet) anchoring, but for weaker anchoring strength all defects will occur on the boundary. These defects will each carry a fractional winding number; such boundary defects are known as "boojums". The boojums will occur in ordered pairs along the boundary; for nematic director with angle oblique to the normal vector, they serve to reduce the winding of the phase in two steps, in order to avoid the formation of interior defects. We determine the number and location of the defects via a Renormalized Energy and numerical simulations.

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arXiv2020

Mass splitting in the Thomas-Fermi-Dirac-von Weizsäcker model with background potential

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 and which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants, and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations.

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arXiv2019

Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

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. Here we consider a limit in which two species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a two-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. We provide a sharp rigorous derivation of the asymptotic limit, both for minimizers and in the context of Gamma-convergence. Geometrical descriptions of limit configurations are derived; among other results, we will show that, quite unexpectedly, coexistence of single and double bubbles can arise. The main difficulties are hidden in the optimal solution of two-component isoperimetric problem: compared to binary systems, not only it lacks an explicit formula, but, more crucially, it can be neither concave nor convex on parts of its domain.

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arXiv2018

Droplet phase in a nonlocal isoperimetric problem under confinement

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 $η$ to represent the size of the domains of the minority phase, and study the resulting droplet regime as $η\to 0$. By considering confinement densities which are spatially variable and attain a nondegenerate maximum, we present a two-stage asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter $M$ which gives the total volume of the droplets at order $η^3$ and its relation to existence and non-existence of Gamow's Liquid Drop model on $\mathbb{R}^3$. For large values of $M$, the minority phase splits into several droplets at an intermediate scale $η^{1/3}$, while for small $M$ minimizers form a single droplet converging to the maximum of the confinement density.

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arXiv2018

Droplet breakup in the liquid drop model with background potential

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 $Z$ and characterize the structure of minimizers in the limit $Z\to 0$ by means of a sharp asymptotic expansion of the energy. In the process of studying this limit we characterize all minimizing sequences for the Gamow model in terms of "generalized minimizers".

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arXiv2017

Spherical particle in a nematic liquid crystal under an external field: the Saturn ring regime

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 field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

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arXiv2017

Ground-states for the liquid drop and TFDW models with long-range attraction

We prove that both the liquid drop model in $\mathbb{R}^3$ with an attractive background nucleus and the Thomas-Fermi-Dirac-von Weizsäcker (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 large $|x|$.) For the TFDW model we adapt classical concentration-compactness arguments by Lions, whereas for the liquid drop model with background attraction we utilize a recent compactness result for sets of finite perimeter by Frank and Lieb.

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arXiv2016

Sharp interface limit of an energy modelling nanoparticle-polymer blends

We identify the $Γ$-limit of a nanoparticle-polymer 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 the density distribution of the infinitely many small nanoparticles. We prove that local minimizers of the limiting energy admit regular phase boundaries and derive necessary conditions of local minimality via the first variation. Finally we discuss possible critical and minimizing patterns in two dimensions and how these patterns vary from global minimizers of the purely local isoperimetric problem.

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arXiv2015

A Degenerate Isoperimetric Problem and Traveling Waves to a Bi-stable Hamiltonian System

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 isoperimetric problem, we identify these isoperimetric curves, appropriately parametrized, as traveling wave solutions to a bi-stable Hamiltonian system of PDE's. We also determine the existence of a maximal propagation speed for these traveling waves through an explicit upper bound depending on the conformal factor.

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arXiv2015

Minimizers of the Landau-de Gennes energy around a spherical colloid particle

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 of radius r_0, satisfy homeotropic weak anchoring at the surface of the colloid, and approach a uniform uniaxial state at infinity. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale r_0< >L. The relationship between the radius and the anchoring strength W 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.

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arXiv2014

Vortex structure in p-wave superconductors

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 and coupled in their second derivative terms. We prove the existence of energy minimizing solutions in bounded domains $Ω\subset\mathbb R^2$, and consider the existence and qualitative properties (such as the asymptotic behavior) of equivariant solutions defined in all of $\mathbb R^2$. The coupling of the equations at highest order changes the nature of the solutions, and many of the usual properties of classical Ginzburg-Landau vortices either do not hold for the p-wave solutions or are not immediately evident.

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