Panayotis Smyrnelis

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.

Entire Solutions with Six-fold Junctions to Elliptic Gradient Systems with Triangle Symmetry

Abstract We prove the existence of solutions to three-fold symmetric elliptic systems in ℝ2 which have six-fold symmetry, asymptotically approaching each of three minima of the potential as |x| → ∞ in two antipodal sectors of angle π/3.

The harmonic map problem with mixed boundary conditions

Given two polygons S ⊂ R2 and Σ ⊂ Rm with the same number of sides, we prove the existence and uniqueness of a smooth harmonic map u : S → Rm satisfying the mixed boundary conditions for S and Σ. This solution is constructed and characterized as a minimizer of the Dirichlet’s energy in the class of maps which satisfy the first mixed boundary condition. Several properties of the solution are established. We also discuss the mixed boundary conditions for harmonic maps defined in smooth domains of the plane.

Symmetry Breaking and Restoration in the Ginzburg–Landau Model of Nematic Liquid Crystals

In this paper we study qualitative properties of global minimizers of the Ginzburg–Landau energy which describes light–matter interaction in the theory of nematic liquid crystals near the Fréedericksz transition. This model depends on two parameters:

Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

\epsilon >0

u\'\'=\\nabla W(u)

ϵ>0 which is small and represents the coherence scale of the system and

Multiphase Solutions to the Vector Allen–Cahn Equation: Crystalline and Other Complex Symmetric Structures

a\ge 0