arXiv subjects

Patricia Bauman

Publications and source records attributed to Patricia Bauman.

4 recordsLinked to original sources

arXiv2021

Regularity of Minimizers for a General Class of Constrained Energies in Two-Dimensional Domains with Applications to Liquid Crystals

We investigate minimizers defined on a bounded domain $Ω$ in $\mathbb{R}^2$ for singular constrained energy functionals that include Ball and Majumdar's modification of the Landau-de Gennes Q-tensor model for nematic liquid crystals. We prove regularity of minimizers with finite energy and show that their range on compact subdomains of $Ω$ does not intersect the boundary of the constraining set.

math.AP↗

arXiv2018

Analysis of Minimizers of the Lawrence-Doniach Energy for Superconductors in Applied Fields

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder, $ Ω\times[0,L]$, in $\mathbb{R}^3$, where $Ω$ is a bounded simply connected Lipschitz domain in $\mathbb{R}^2$. For an applied magnetic field $\vec{H}_{ex}=h_{ex}\vec{e}_{3}$ that is perpendicular to the layers with $\left|\lnε\right|\ll h_{ex}\llε^{-2}$ as $ε\rightarrow 0$, where $ε$ is the reciprocal of the Ginzburg-Landau parameter, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as $ε$ and the interlayer distance $s$ tend to zero. Under appropriate assumptions on $s$ versus $ε$, we establish comparison results between the minimum Lawrence-Doniach energy and the minimum three-dimensional anisotropic Ginzburg-Landau energy. As a consequence, our asymptotic formula also describes the minimum three-dimensional anisotropic energy as $ε$ tends to zero.

math.AP↗

arXiv2017

Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<ε\le 1$, let $Ω\subset\mathbb R^n$ be a bounded smooth domain and $u_ε:Ω\subset\R^n\to \mathbb R^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\begin{cases} -Δu_ε=\frac{1}{ε^2}(1-|u_ε|^2)u_ε&\ {\rm{in}}\ \ Ω, \frac{\partial u_ε}{\partialν}+λ_ε(u_ε-g_ε)=0 & \ {\rm{on}}\ \ \partialΩ, \end{cases} $$ where the anchoring strength parameter $λ_ε=Kε^{-α}$ for some $K>0$ and $α\in [0,1)$, and $g_ε\in C^2(\partialΩ, \mathbb S^1)$. Motivated by the connection with the Landau-De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the {asymptotic behavior} of $u_ε$ as $ε$ goes to zero under the condition that the total modified Ginzburg-Landau energy {satisfies} $F_ε(u_ε,Ω)\le M|\logε|$ for some $M>0$.

math.AP↗

arXiv2015

Regularity and the Behavior of Eigenvalues for Minimizers of a Constrained $Q$-tensor Energy for Liquid Crystals

We investigate minimizers defined on a bounded domain in $\mathbb{R}^2$ for the Maier--Saupe Q--tensor 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 have eigenvalues 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 have eigenvalues strictly within the physical range.

math.AP↗