Volodymyr Rybalko

Minimax Critical Points in Ginzburg-Landau Problems with Semi-stiff Boundary Conditions: Existence and Bubbling

Let Ω ⊂ ℝ2 be a smooth bounded simply connected domain. We consider the simplified Ginzburg-Landau energy , where u: Ω → ℂ. We prescribe |u| = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E ϵ. Using a mountain pass approach, we obtain existence of critical points of E ϵ for large ϵ. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.

On an evolution equation in a cell motility model

Abstract This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction–diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth ( H 2 ) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solutions exist rely on a maximum principle type argument. We also explore the long time behavior of the model using both analytical and numerical tools. We prove the nonexistence of traveling wave solutions with nonzero velocity. Numerical experiments show that presence of non-linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion. This is done by developing an algorithm for efficient numerical resolution of the non-local term in the evolution equation.

On the first eigenpair of singularly perturbed operators with oscillating coefficients

ABSTRACT The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behavior of the first eigenpair (ground state) of this problem. The main tool in deriving the limit \mbox(effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.

Vortex phase separation in mesoscopic superconductors

We demonstrate that in mesoscopic type II superconductors with the lateral size commensurate with London penetration depth, the ground state of vortices pinned by homogeneously distributed columnar defects can form a hierarchical nested domain structure. Each domain is characterized by an average number of vortices trapped at a single pinning site within a given domain. Our study marks a radical departure from the current understanding of the ground state in disordered macroscopic systems and provides an insight into the interplay between disorder, vortex-vortex interaction, and confinement within finite system size. The observed vortex phase segregation implies the existence of the soliton solution for the vortex density in the finite superconductors and establishes a new class of nonlinear systems that exhibit the soliton phenomenon.

Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes

We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the

Renormalized Ginzburg-Landau energy and location of near boundary vortices

\Gamma

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes.

Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain

We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.