23,347 research outputs found
Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure
In this paper we are concerned with convergence of solutions of the Poisson
equation with Neumann boundary conditions in a two-dimensional thin domain
exhibiting highly oscillatory behavior in part of its boundary. We deal with
the resonant case in which the height, amplitude and period of the oscillations
are all of the same order which is given by a small parameter .
Applying an appropriate corrector approach we get strong convergence when we
replace the original solutions by a kind of first-order expansion through the
Multiple-Scale Method.Comment: to appear in Quarterly of Applied Mathematic
Magnetic monopole and string excitations in a two-dimensional spin ice
We study the magnetic excitations of a square lattice spin-ice recently
produced in an artificial form, as an array of nanoscale magnets. Our analysis,
based upon the dipolar interaction between the nanomagnetic islands, correctly
reproduces the ground-state observed experimentally. In addition, we find
magnetic monopole-like excitations effectively interacting by means of the
usual Coulombic plus a linear confining potential, the latter being related to
a string-like excitation binding the monopoles pairs, what indicates that the
fractionalization of magnetic dipoles may not be so easy in two dimensions.
These findings contrast this material with the three-dimensional analogue,
where such monopoles experience only the Coulombic interaction. We discuss,
however, two entropic effects that affect the monopole interactions: firstly,
the string configurational entropy may loose the string tension and then, free
magnetic monopoles should also be found in lower dimensional spin ices;
secondly, in contrast to the string configurational entropy, an entropically
driven Coulomb force, which increases with temperature, has the opposite effect
of confining the magnetic defects.Comment: 8 pages. Accepted by Journal of Applied Physics (2009
Error estimates for a Neumann problem in highly oscillating thin domains
In this work we analyze convergence of solutions for the Laplace operator
with Neumann boundary conditions in a two-dimensional highly oscillating domain
which degenerates into a segment (thin domains) of the real line. We consider
the case where the height of the thin domain, amplitude and period of the
oscillations are all of the same order, given by a small parameter .
We investigate strong convergence properties of the solutions using an
appropriate corrector approach. We also give error estimates when we replace
the original solutions for the second-order expansion through the
Multiple-Scale Method
- …