735 research outputs found
On the heteroclinic connection problem for multi-well gradient systems
We revisit the existence problem of heteroclinic connections in
associated with Hamiltonian systems involving potentials
having several global minima. Under very mild
assumptions on 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 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 , and represents a more geometric
alternative to the approaches for finding such connections described, for
example, by N.D. Alikakos and G.Fusco in , by S.V. Bolotin in
, by J. Byeon, P.
Montecchiari, and P. Rabinowitz in
, and by P. Rabinowitz in .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 .
Here P_{\Omega_\e}(\{u(x)=1\}) denotes the perimeter of the set
in \Omega_\e, \fint denotes the integral average and 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 as 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 -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 to the
elliptic system where is a `triple-well' potential. This solution is a local minimizer of
the associated energy in the sense
that minimizes the energy on any compact set among competitors agreeing
with outside that set. Furthermore, we show that along subsequences, the
`blowdowns' of given by approach a minimal triple junction
as . 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
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