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 $\texttt{Homoclinic and heteroclinic orbits for a class of Hamiltonian}$ $\texttt{systems}$.Comment: 19 pages, 3 figures. KEYWORDS: heteroclinic orbits, multi-well potentials, minimizing geodesic

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For \Omega_\e=(0,\e)\times (0,1) a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \inf_u E^{\gamma}_{\Omega_\e}(u) where E^{\gamma}_{\Omega_\e}(u):= P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx and the minimization is taken over competitors u\in BV(\Omega_\e;\{\pm 1\}) satisfying a mass constraint \fint_{\Omega_\e}u=m for some $m\in (-1,1)$. Here P_{\Omega_\e}(\{u(x)=1\}) denotes the perimeter of the set $\{u(x)=1\}$ in \Omega_\e, \fint denotes the integral average and $v$ denotes the solution to the Poisson problem -\Delta v=u-m\;\mbox{in}\;\Omega_\e,\quad\nabla v\cdot n_{\partial\Omega_\e}=0\;\mbox{on}\;\partial\Omega_\e,\quad\int_{\Omega_\e}v=0. We show that a striped pattern is the minimizer for \e\ll 1 with the number of stripes growing like $\gamma^{1/3}$ as $\gamma\to\infty.$ We then present generalizations of this result to higher dimensions.Comment: 20 pages, 2 figure

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

We use the method of $\Gamma$-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

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

Allen-Cahn Solutions with Triple Junction Structure at Infinity

We construct an entire solution $U:\mathbb{R}^2\to\mathbb{R}^2$ to the elliptic system $$\Delta U=\nabla_uW(U),$$ where $W:\mathbb{R}^2\to [0,\infty)$ is a `triple-well' potential. This solution is a local minimizer of the associated energy $$\int \frac{1}{2}|\nabla U|^2+W(U)\,dx$$ in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set. Furthermore, we show that along subsequences, the `blowdowns' of $U$ given by $U_R(x):=U(Rx)$ approach a minimal triple junction as $R\to\infty$. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions