# Panayotis SmyrnelisNational and Kapodistrian University of Athens | uoa

Panayotis Smyrnelis

PhD

## About

36

Publications

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153

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Citations since 2016

Introduction

## Publications

Publications (36)

The anisotropic Ginzburg–Landau system $$\begin{aligned} \Delta u+\delta \nabla ({{\,\mathrm{div}\,}}u) +\delta {{\,\mathrm{curl}\,}}^*({{\,\mathrm{curl}\,}}u)=(|u|^2-1) u, \end{aligned}$$for \(u:\mathbb R^2\rightarrow \mathbb R^2\) and \(\delta \in (-1,1)\), models the formation of vortices in liquid crystals. We prove the existence of entire solu...

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescr...

In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential $W:\mathbb{R}^m\to\mathbb{R}$, $m\geq 2$, there exis...

We construct double layered solutions to the extended Fisher–Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher–Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these s...

In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential W:Rm→R, m≥2, there exists an arbitrary small perturb...

The extended Painlevé P.D.E. system Δy−x1y−2|y|2y=0, (x1,…,xn)∈Rn, y:Rn→Rm, is obtained by multiplying by −x1 the linear term of the Ginzburg-Landau equation Δη=|η|2η−η, η:Rn→Rm. The two dimensional model n=m=2 describes in the theory of light-matter interaction in liquid crystals, the orientation of the molecules at the boundary of the illuminated...

We establish a general comparison principle for vector valued minimizers of a functional, associated when the potential is smooth, to elliptic gradient systems. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential. In the scalar case, this conditi...

We construct double layered solutions to the extended Fisher-Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher-Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these s...

The second Painlev\'e O.D.E. $y''-xy-2y^3=0$, $x\in \mathbb{R},$ has been extensively studied since the early 1900's, due to its importance for applications. Recently, we discovered that the extended equation: $\Delta y -x_1 y - 2 |y|^2y=0$, $(x_1,\ldots,x_n)\in \mathbb{R}^n$, $y:\mathbb{R}^n\to\mathbb{R}^m$, is relevant in the theory of light-matt...

Optical vortices and lattices of these are attracting the attention of the scientific community because of their applications in various fields of optical processing, communications, enhanced imaging systems, and bio-inspired devices. Programmable optical vortices lattices with arbitrary distributions have been achieved using illuminated liquid cry...

We prove a general theorem on the existence of heteroclinic orbits in Hilbert spaces, and present a method to reduce the solutions of some P.D.E. problems to such orbits. In our first application, we give a new proof in a slightly more general setting of the heteroclinic double layers (initially constructed by Schatzman), since this result is parti...

In this paper we describe domain walls appearing in a thin, nematic liquid crystal sample subject to an external field with intensity close to the Fr\'eedericksz transition threshold. Using the gradient theory of the phase transition adopted to this situation, we show that depending on the parameters of the system, domain walls occur in the bistabl...

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 Friedrichs transition. This model is depends on two parameters: $\epsilon>0$ which is small and represents the coherence scale of the system and $a\geq 0$ which r...

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 magnet...

This chapter together with Chap. 4 contain some general tools for obtaining estimates for systems.

In this chapter we extend the density estimate in Theorem 5.2 by replacing the constant solution a with a symmetric, minimal, hyperbolic connection e, (and more generally with any equivariant minimal hyperbolic solution), and then derive Liouville theorems and asymptotic information for minimal solutions under symmetry hypotheses. Utilizing the ext...

The main object in this chapter is the stress-energy tensor, which is an algebraic fact implying several useful identities like the (weak) monotonicity formula, Gui’s Hamiltonian identities, and Pohozaev’ identities, for all solutions and all potentials W ≥ 0. Modica’s inequality holds in the scalar case and implies a strong monotonicity formula, b...

We present a systematic study of entire symmetric solutions u: ℝⁿ→ ℝm of the vector Allen–Cahn equation Δu − Wu(u) = 0, x∈ ℝⁿ, where W: ℝm→ ℝ is smooth, symmetric, nonnegative with a finite number of zeros, and Wu:= (∂W∕∂u1, …, ∂W∕∂um)⊤. We assume that W is invariant under a finite reflection group Γ acting on target space ℝm and that there is a fi...

In this chapter we begin the study of entire solutions \(u:{\mathbb R}^n\rightarrow {\mathbb R}^n\) of the vector Allen–Cahn equation (6.1) that describe the coexistence of different phases in a neighborhood of a point. We work in a symmetry context where a finite reflection group G is acting both on the domain space \({\mathbb R}_x^n\) and on the...

We begin by giving a concise proof of the existence of a heteroclinic connection (Theorem 2.1). The experienced reader then can move on to Sect. 2.6. In Sect. 2.4 we develop an alternative approach via constrained minimization. Most readers will find this easier and also good preparation for the polar form and the cut-off lemma in Chap. 4. In Sect....

Let \(W:{{\mathbb R}}^m\rightarrow {{\mathbb R}}\) be a nonnegative potential with exactly two nondegenerate zeros \(a^-\neq a^+\in {{\mathbb R}}^m\). Assume that there are N ≥ 1 distinct heteroclinic orbits connecting a⁻ to a⁺, represented by maps \(\bar {u}_1,\ldots ,\bar {u}_N\) that minimize the one-dimensional energy \(J_{{\mathbb R}}(u)=\int...

In this chapter we establish a maximum principle type result that provides pointwise control on minimal solutions. In contrast to the usual maximum principle, it does not hold for solutions in general, not even for local minimizers in the scalar case. We obtain it as a corollary of a replacement lemma modeled after Lemmas 2.4 and 2.5.

This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes – non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two pha...

We prove the existence of minimal heteroclinic orbits for a class of fourth order O.D.E. systems with variational structure. In our general set-up, the set of equilibria of these systems is a union of manifolds, and the heteroclinic orbits connect two disjoint components of this set.

We present a systematic study of entire symmetric solutions $u:R^n\rightarrow
R^m$ of the vector Allen-Cahn equation $\Delta u-W_u(u)=0, x \in R^n$, where
$W:R^m\rightarrow R$ is smooth, symmetric, nonnegative with a finite number of
zeros and $W_u=(\frac{\partial W}{\partial u_1},\ldots,\frac{\partial
W}{\partial u_m})^\top$. We introduce a genera...

We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the {\it shadow kink}. Its loca...

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per a 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 magn...

A maximum principle is established for minimal solutions to the system δu-∇W(u)=0, with a potential W vanishing at the boundary of a closed convex set C0⊂Rm, which is either C2 smooth or coincides with a point (a).

In the first part of the paper, we establish two necessary conditions for the existence of bounded one-dimensional minimizers u: the potential W must have a global minimum supposed to be 0 without loss of generality, and W(u(x)) → 0 as |x| → ∞. Furthermore, non-constant minimizers connect at ± ∞ two distinct components of the set {W = 0}. In the se...

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 conditi...

A periodic connection is constructed for a double well potential defined in
the plane. This solution invalidates Modica's estimate as well as the
corresponding Liouville Theorem for general phase transition potentials.
Gradients estimates are also established for several kind of elliptic systems.
They allow us to prove in some particular cases the...

We prove the existence of solutions to three-fold symmetric elliptic systems in $\R^2$ which have six-fold symmetry, asymptotically approaching each of three minima of the potential as $|x|\to\infty$ in two antipodal sectors of angle $\pi/3$.

Under the assumption that the potential W is invariant under a general discrete reflection group G = T G acting on R n , we establish existence of G -equivariant solutions to ∆u − Wu(u) = 0, and find an estimate. By taking the size of the cell of the lattice in space domain to infinity, we obtain that these solutions converge to G-equivariant solut...