2015-08-17

Regularity of Minimizers to a Constrained Q-tensor Energy for Liquid CrystalsWe investigate minimizers defined on a bounded domain in for the Maier--Saupe energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification of the Ginzburg-Landau 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.

Sawtooth profile of smectic A liquid crystals

We study de Gennes free energy for smectic A liquid crystals over valued vector fields to understand the chevron (zigzag) pattern formed in the presence of an applied magnetic field. Well above the instability threshold, we show via -convergence that a chevron structure where the director connects two minimum states of the sphere is favored. Numerical simulations illustrating the chevron structures for the de Gennes energy will be presented.

Defects in nematic colloids

Nematic liquid crystals are characterized by their long-range orientational order: the molecules tend to align in a common direction.
When foreign particles are immersed, the alignment is distorted, creating topological defects and fascinating self-assembly phenomena.
In a joint work with S. Alama and L. Bronsard we study the nematic structure induced by one spherical particle, using (the Ginzburg-Landau-like) Landau-de Gennes theory. Depending on particle size and surface anchoring, we give accurate descriptions of either quadrupolar configurations with Saturn-ring defect, or dipolar configurations with single hyperbolic defect.

2015-08-18

Analysis of Energy Minimizers for Layered Superconductors in Three Dimensions in Intermediate Applied Magnetic FieldsWe analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder, , in three dimensions. In an applied magnetic field that is perpendicular to the horizontal superconducting layers and has modulus h satisfying ||<< h < < (1/epsilon)^2 as epsilon tends to zero, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as epsilon and the interlayers distance s tend to zero. We also establish comparison results between the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropic Ginzburg-Landau anisotropic energy for a superconductor occupying the cylinder.

Radial mixed type solutions in Chern-Simons gauge theories

Abstract Gauge theories in (2+1)-dimensional space has been investigated in recent decades not only to explain certain physical phenomena such as superconductivity but also to find all solitonic stuructures for future use.
Especially, gauge theories with Chern-Simons term exhibit exotic objects such as nontopological vortex and mixed type solutions. Mixed type solutions appears only in nonAbelian gauge theories.
In this talk, we present some recent development on mixed type solutions in rank 2 CS gauge theories

The linearized problem of nonlinear elliptic equations near the critical exponents.

In this talk we consider some qualitative characteristics of multi-bubble solutions to the Lane-Emden-Fowler equations with slightly subcritical exponents given any dimension . By examining the linearized problem at each m-bubble solution, we provide a number of estimates on the first -eigenvalues and their corresponding eigenfunctions. Specifically, we present a new proof of the classical theorem due to Bahri-Li-Rey (Calc. Var. Partial Differential Equations 3 (1995) 67-93) which states that if , then the Morse index of a multi-bubble solution is governed by a certain symmetric matrix whose component consists of a combination of Green's function, the Robin function, and their first and second derivatives. Our proof also allows us to handle the intricate case . This is a joint work with Seunghyeok Kim and Ki-Ahm Lee.

A "permeability" defined through singular perturbations.

Determining the permeability of membrane appears as a crucial issue in many branches of mathematical sciences.
However, the one widely employed contains ``ambiguity" which may create a serious problem. In this talk, I will propose a concept which is closely related to the concept of the permeability based on the idea of singular perturbation.

Asymtotic self-similarity of positive entire solutions for quasilinear elliptic equations of Lane-Emden type

Asymptotic behavior is a tool to classify positive entire solutions of nonlinear elliptic equations.
However, this issue is not appreciated properly yet for quasilinear elliptic equations of Lane-Emden type. In this talk, I will discuss the differences depending the p-laplace operator and the exponent describing the nonlinearity. Main concepts are asymptotic self-similarity and separation.

Existence and Regularity of Solutions to Nonlinear Maxwell Equations

We examine regularity of weak solutions of several nonlinear Maxwell systems by using of the reduction method. This method reduces the original system into two div-curl systems and an oblique derivative problem of a quasilinear elliptic equation, and makes it possible to improve the regularity of the solutions by iteration. The reduction method is also used to show existence of steady states of a thermoelectrical model.

Phase-field model of dynamical interfaces

We talk about a phase-field model. The model is based on the decrease of the free energy of the system, in which the order parameter varies from -1 to 1. This diffuse-interface formulation gives us an alternative way to approximate the sharp interface models such as mean curvature flow, volume-preserving mean curvature flow, and surface diffusion flow. In this talk, we observe the motions driven by the phase-field model and how they evolve from convex and nonconvex curves. Especially, the Allen-Cahn equation will be considered with and without boundary conditions.

2015-08-19

Unbounded solutions for periodic nonlinear elliptic problemsFor nonlinear elliptic problems with periodic nonlinearities, there is a rich variety of solutions which are bounded or unbounded.
For bounded solutions, it is related to the Allen-Cahn equation; for unbounded solutions, to a work of Moser which is a PDE version of Aubry-Mather theory in Hamiltonian systems. I would like to introduce my recent works with Paul Rabinowitz for construction of new types of bounded solutions and unbounded solutions.

Maxwell-Chern-Simons vortices on compact surfaces

In this talk, we study the Maxwell-Chern-Simons-Higgs vortices on a compact Riemann surface.
We establish the existence of a solution of the static nonself-dual Maxwell-Chern-Simons-Higgs vortex equations, which is a minimizer of the static energy functional.
This shows the nonequivalence of the first and the second order Maxwell-Chern-Simons-Higgs vortex equations.
The nonequivalence is also proved for the Chern-Simons-Higgs vortices by verifying the Chern-Simons limit.

Global solutions to time-dependent Ginzburg-Landau-Chern-Simons equations

We propose and study a time-dependent Ginzburg-Landau-Chern-Simons model.
The local existence of solutions to the model in a bounded domain $\Omega$ is proved by applying contraction mapping theorem. We use covariant energy estimate to obtain global solutions.