arXiv subjects

Dominik Stantejsky

Publications and source records attributed to Dominik Stantejsky.

11 recordsLinked to original sources

arXiv2025

A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals

Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional surface $T$ and the length of its boundary $\partial T$ reduced by a prescribed curve to make our problem non-trivial. We additionally include an obstacle $E$ for $T$ and pose a surface energy on $E$. We present an algorithm based on the Alternating Direction Method of Multipliers that minimizes a discretized version of the energy using finite elements, generalizing existing TV-minimization methods. We study different inclusion shapes demonstrating the rich structure of minimizing configurations and provide physical interpretation of our findings for colloidal particles in nematic liquid crystal.

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arXiv2025

A priori estimates and $η-$compactness for anisotropic Ginzburg-Landau minimizers with tangential anchoring

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence or curl penalization on a simply-connected two-dimensional domain $Ω$. On the boundary, strong tangential anchoring is imposed. We prove a priori estimates for $u_\varepsilon$ in $L^\infty$ uniform in $\varepsilon$ and that the Lipschitz constant of $u_\varepsilon$ blows up like $\varepsilon^{-1}$. We then deduce compactness for a subsequence that converges to an $\mathbb{S}^1-$valued map with either one interior point defect or two boundary half-defects. We conclude our study with a proof that no boundary vortices can occur in the divergence penalized case.

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arXiv2025

Asymptotics for Minimizers of Landau-de Gennes with a Magnetic Field and Tangential Anchoring

In this article we prove existence of minimizers of the Landau-de Gennes energy for liquid crystals with homogeneous external magnetic field and strong uniaxial planar anchoring. Next we consider the asymptotics of solutions to the joint minimization of the energy w.r.t. the function and its boundary condition. This constitutes a generalization to arbitrary regular particle shapes of the results obtained in [BLS, arXiv:2403.20274] in a particular setting. Moreover, we show the absence of line singularities in some asymptotic parameter regime. Finally we characterize the optimal orientation of particles vis-à-vis the magnetic field direction and compute it explicitly for different particle shapes.

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arXiv2025

Interaction energies in nematic liquid crystal suspensions

We establish, as $ρ\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $ρ$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where $μ_j$ is the minimal energy after zooming in at scale $ρ$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.

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arXiv2025

Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring

Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of $\mathbb{S}^2-$valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in $\mathbb{R}^{3}$. In particular, we obtain a monotonicity formula respecting tangentiality on a curved boundary in order to show optimal regularity up to the boundary. We introduce novel sufficient conditions under which the minimizer must exhibit symmetries. Under a symmetry assumption, we present a delineation of the singularities of minimizers, namely that a mimimizer has exactly two point singularities, located on the boundary at opposite points.

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arXiv2024

Spherical Particle in Nematic Liquid Crystal with a Magnetic Field and Planar Anchoring

We study minimizers of the Landau-de Gennes energy in $\mathbb{R}^3\setminus B_1(0)$ with external magnetic field in the large particle limit. We impose strong tangential anchoring and uniaxiality of the $Q-$tensor on the boundary. We derive a lower bound for the energy in terms of the boundary condition and show in the extreme cases of strong and weak magnetic field strength that the longitudinal director field is energy minimizing, indicating the presence of two half-point defects, so called boojums, at two opposite points of the sphere. Using a recovery sequence, we show that the energy bound is optimal in these extreme cases.

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arXiv2024

On a Divergence Penalized Landau-de Gennes Model

We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a uniform uniaxial state as $|x|\to\infty$. We study the "small particle" limit and obtain a representation formula for solutions to the associated Euler-Lagrange equations. We also present a numerical analysis of these equations based on a finite element approach and discuss the effect of the divergence penalization on the "Saturn ring" defects and on the properties of the $Q$-tensor.

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arXiv2024

Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

We use the Landau-de Gennes energy to describe a particle immersed into nematic liquid crystals with a constant applied magnetic field. We derive a limit energy in a regime where both line and point defects are present, showing quantitatively that the close-to-minimal energy is asymptotically concentrated on lines and surfaces nearby or on the particle. We also discuss regularity of minimizers and optimality conditions for the limit energy.

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arXiv2024

A priori $L^\infty-$bound for Ginzburg-Landau energy minimizers with divergence penalization

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain $Ω$. On the boundary, strong tangential anchoring is imposed. We prove that minimizers satisfy a $L^\infty$-bound uniform in $\varepsilon$ when $Ω$ has $C^{2,1}-$boundary and that the Lipschitz constant blows up like $\varepsilon^{-1}$ when $Ω$ has $C^{3,1}-$boundary. Our theorem extends to $W^{2,p}-$regularity result for our elliptic system with mixed Dirichlet-Neumann boundary condition.

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arXiv2022

Asymptotic shape of isolated magnetic domains

We investigate the energy of an isolated magnetized domain $Ω\subset \mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given by $$ \mathcal{E}(Ω) \ = \ \int_{\mathbb{R}^n} |\nabla χ_Ω| \ dx + \int_{\mathbb{R}^n} |\nabla h_Ω|^2 \ dx $$ penalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential $h_Ω$ is determined by $Δh_Ω= \partial_1 χ_Ω$, corresponding to uniform magnetization within the domain. We consider the macroscopic regime $|Ω| \rightarrow \infty$, in which we derive compactness and $Γ$-limit which is formulated in terms of the cross-sectional area of the anisotropically rescaled configuration. We then give the solutions for the limit problems.

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arXiv2020

The saturn ring effect in nematic liquid crystals with external field: effective energy and hysteresis

In this work we consider the Landau-de Gennes model for liquid crystals with an external electromagnetic field to model the occurrence of the saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and saturn ring configuration and the occurrence of a hysteresis phenomenon, giving a rigorous explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B 10, 311-321 (1999)].

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