arXiv subjects

Peter Sternberg

Publications and source records attributed to Peter Sternberg.

27 records · Page 2Linked to original sources

arXiv2017

Dimension reduction for the Landau-de Gennes model on curved nematic thin films

We use the method of $Γ$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach that we used previously to study a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustrum.

math.AP↗

arXiv2016

A Degenerate Isoperimetric Problem in the Plane

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the metric ds^2 near the singularities equals a quadratic polynomial times the standard metric. Here we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, appropriately parametrized, arise as traveling wave solutions to a bi-stable Hamiltonian system.

math.DG↗

arXiv2016

On the heteroclinic connection problem for multi-well gradient systems

We revisit the existence problem of heteroclinic connections in $\mathbb{R}^N$ associated with Hamiltonian systems involving potentials $W:\mathbb{R}^N\to \mathbb{R}$ having several global minima. Under very mild assumptions on $W$ we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor $\sqrt{W}.$ Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of P.Sternberg in $\texttt{Vector-valued local minimizers of nonconvex}$ $\texttt{variational problems}$, and represents a more geometric alternative to the approaches for finding such connections described, for example, by N.D. Alikakos and G.Fusco in $\texttt{On the connection problem for potentials with}$ $\texttt{several global minima}$, by S.V. Bolotin in $\texttt{Libration motions of natural dynamical systems}$, by J. Byeon, P. Montecchiari, and P. Rabinowitz in $\texttt{A double well potential}$ $\texttt{system}$, and by P. Rabinowitz in $\texttt{Homoclinic and heteroclinic orbits for a class of Hamiltonian}$ $\texttt{systems}$.

math.AP↗

arXiv2015

A Degenerate Isoperimetric Problem and Traveling Waves to a Bi-stable Hamiltonian System

We analyze a non-standard isoperimetric problem in the plane associated with a metric having degenerate conformal factor at two points. Under certain assumptions on the conformal factor, we establish the existence of curves of least length under a constraint associated with enclosed Euclidean area. As a motivation for and application of this isoperimetric problem, we identify these isoperimetric curves, appropriately parametrized, as traveling wave solutions to a bi-stable Hamiltonian system of PDE's. We also determine the existence of a maximal propagation speed for these traveling waves through an explicit upper bound depending on the conformal factor.

math.AP↗

arXiv2015

Dimension reduction for the Landau-de Gennes model in planar nematic thin films

We use the method of $Γ$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic $Q$-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.

math.AP↗

arXiv2014

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For $Ω_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^γ_{Ω_\e}(u)\] where \[ E^γ_{Ω_\e}(u):= P_{Ω_\e}(\{u(x)=1\})+γ\int_{Ω_\e}\abs{\nabla{v}}^2\,dx \] and the minimization is taken over competitors $u\in BV(Ω_\e;\{\pm 1\})$ satisfying a mass constraint $\fint_{Ω_\e}u=m$ for some $m\in (-1,1)$. Here $P_{Ω_\e}(\{u(x)=1\})$ denotes the perimeter of the set $\{u(x)=1\}$ in $Ω_\e$, $\fint$ denotes the integral average and $v$ denotes the solution to the Poisson problem \[ -Δv=u-m\;\mbox{in}\;Ω_\e,\quad\nabla v\cdot n_{\partialΩ_\e}=0\;\mbox{on}\;\partialΩ_\e,\quad\int_{Ω_\e}v=0.\] We show that a striped pattern is the minimizer for $\e\ll 1$ with the number of stripes growing like $γ^{1/3}$ as $γ\to\infty.$ We then present generalizations of this result to higher dimensions.

math.AP↗

arXiv2013

Kinematic and dynamic vortices in a thin film driven by an applied current and magnetic field

Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular thin film superconductor subjected to an applied current fed into a portion of the sides and an applied magnetic field directed orthogonal to the film. Through a center manifold reduction we develop a rigorous bifurcation theory for the appearance of periodic solutions in certain parameter regimes near the normal state. The leading order dynamics yield in particular a motion law for kinematic vortices moving up and down the center line of the sample. We also present computations that reveal the co-existence and periodic evolution of kinematic and magnetic vortices.

math.AP↗

arXiv2012

Dynamics Of Ginzburg-Landau And Gross-Pitaevskii Vortices On Manifolds

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy identities.

math.DS↗

arXiv2007

The resistive state in a superconducting wire: Bifurcation from the normal state

We study formally and rigorously the bifurcation to steady and time-periodic states in a model for a thin superconducting wire in the presence of an imposed current. Exploiting the PT-symmetry of the equations at both the linearized and nonlinear levels, and taking advantage of the collision of real eigenvalues leading to complex spectrum, we obtain explicit asymptotic formulas for the stationary solutions, for the amplitude and period of the bifurcating periodic solutions and for the location of their zeros or "phase slip centers" as they are known in the physics literature. In so doing, we construct a center manifold for the flow and give a complete description of the associated finite-dimensional dynamics.

math-ph↗