64 research outputs found
On well-posedness of variational models of charged drops
Electrified liquids are well known to be prone to a variety of interfacial
instabilities that result in the onset of apparent interfacial singularities
and liquid fragmentation. In the case of electrically conducting liquids, one
of the basic models describing the equilibrium interfacial configurations and
the onset of instability assumes the liquid to be equipotential and interprets
those configurations as local minimizers of the energy consisting of the sum of
the surface energy and the electrostatic energy. Here we show that,
surprisingly, this classical geometric variational model is mathematically
ill-posed irrespectively of the degree to which the liquid is electrified.
Specifically, we demonstrate that an isolated spherical droplet is never a
local minimizer, no matter how small is the total charge on the droplet, since
the energy can always be lowered by a smooth, arbitrarily small distortion of
the droplet's surface. This is in sharp contrast with the experimental
observations that a critical amount of charge is needed in order to destabilize
a spherical droplet. We discuss several possible regularization mechanisms for
the considered free boundary problem and argue that well-posedness can be
restored by the inclusion of the entropic effects resulting in finite screening
of free charges.Comment: 18 pages, 2 figure
Asymptotic properties of ground states of scalar field equations with a vanishing parameter
We study the leading order behaviour of positive solutions of the equation
-\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where , and when is a small parameter. We give a complete
characterization of all possible asymptotic regimes as a function of ,
and . The behavior of solutions depends sensitively on whether is less,
equal or bigger than the critical Sobolev exponent . For
the solution asymptotically coincides with the solution of the
equation in which the last term is absent. For the solution
asymptotically coincides with the solution of the equation with
. In the most delicate case the asymptotic behaviour
of the solutions is given by a particular solution of the critical
Emden--Fowler equation, whose choice depends on in a nontrivial
way
Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction
Recent advances in nanofabrication make it possible to produce multilayer
nanostructures composed of ultrathin film materials with thickness down to a
few monolayers of atoms and lateral extent of several tens of nanometers. At
these scales, ferromagnetic materials begin to exhibit unusual properties, such
as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also
referred to as Dzyaloshinskii-Moriya interaction (DMI), because of the
increased importance of interfacial effects. The presence of surface DMI has
been demonstrated to fundamentally alter the structure of domain walls. Here we
use the micromagnetic modeling framework to analyse the existence and structure
of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy
functional. We explicitly construct the minimizers in the one-dimensional
setting, both for the interior and edge walls, for a broad range of parameters.
We then use the methods of {}-convergence to analyze the asymptotics of
the two-dimensional mag- netization patterns in samples of large spatial extent
in the presence of weak applied magnetic fields
Low density phases in a uniformly charged liquid
This paper is concerned with the macroscopic behavior of global energy
minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki
model of diblock copolymer melts. This model is also referred to as the nuclear
liquid drop model in the studies of the structure of highly compressed nuclear
matter found in the crust of neutron stars, and, more broadly, is a paradigm
for energy-driven pattern forming systems in which spatial order arises as a
result of the competition of short-range attractive and long-range repulsive
forces. Here we investigate the large volume behavior of minimizers in the low
volume fraction regime, in which one expects the formation of a periodic
lattice of small droplets of the minority phase in a sea of the majority phase.
Under periodic boundary conditions, we prove that the considered energy
-converges to an energy functional of the limit "homogenized" measure
associated with the minority phase consisting of a local linear term and a
non-local quadratic term mediated by the Coulomb kernel. As a consequence,
asymptotically the mass of the minority phase in a minimizer spreads uniformly
across the domain. Similarly, the energy spreads uniformly across the domain as
well, with the limit energy density minimizing the energy of a single droplets
per unit volume. Finally, we prove that in the macroscopic limit the connected
components of the minimizers have volumes and diameters that are bounded above
and below by universal constants, and that most of them converge to the
minimizers of the energy divided by volume for the whole space problem.Comment: arXiv admin note: text overlap with arXiv:1304.4318 by other author
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