Convergence of phase-field approximations to the Gibbs-Thomson law

We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.Comment: 25 page

Convergence of the Allen-Cahn equation with Neumann boundary conditions

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.Comment: 26 pages, 1 figur

Dimerization-induced enhancement of the spin gap in the quarter-filled two-leg rectangular ladder

We report density-matrix renormalization group calculations of spin gaps in the quarter-filled correlated two-leg rectangular ladder with bond-dimerization along the legs of the ladder. In the small rung-coupling region, dimerization along the leg bonds can lead to large enhancement of the spin gap. Electron-electron interactions further enhance the spin gap, which is nonzero for all values of the rung electron hopping and for arbitrarily small bond-dimerization. Very large spin gaps, as are found experimentally in quarter-filled band organic charge-transfer solids with coupled pairs of quasi-one-dimensional stacks, however, occur within the model only for large dimerization and rung electron hopping that are nearly equal to the hopping along the legs. Coexistence of charge order and spin gap is also possible within the model for not too large intersite Coulomb interaction

On the standing wave mode of giant pulsations

Both odd-mode and even-mode standing wave structures have been proposed for giant pulsations. Unless a conclusion is drawn on the field-aligned mode structure, little progress can be made in understanding the excitation mechanism of giant pulsations. In order to determine the standing wave mode, we have made a systematic survey of magnetic field data from the AMPTE CCE spacecraft and from ground stations located near the geomagnetic foot point of CCE. We selected time intervals when CCE was close to the magnetic equator and also magnetically close to Syowa and stations in Iceland, and when either transverse or compressional Pc 4 waves were observed at CCE. Magnetograms from the ground stations were then examined to determine if there was a giant pulsation in a given time interval. One giant pulsation was associated with a compressional wave, while no giant pulsation was observed in association with transverse wave events. The CCE magnetic field record for the giant pulsation exhibited a remarkable similarity to a giant pulsation observed from the ATS 6 geostationary satellite near the magnetic equator (Hillebrand et al., 1982). In agreement with Hillebrand et al., we conclude that the compressional nature of the giant pulsation is due to an odd-mode standing wave structure. This conclusion places a strong constraint on the generation mechanism of giant pulsations. In particular, if giant pulsations are excited through the drift bounce resonance of ions with standing Alfvén waves, ω - mωd = ±Nωb, where ω is the wave frequency, m is the azimuthal wave number, ωd is the ion drift frequency,N is an integer, and ωb is the ion bounce frequency, then the resonance must occur at an even N

Phase field model with a variable chemical potential

We study some asymptotic behavior of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a control of the total energy and chemical potential. We show that the limit interface is an integral varifold with generalized LP mean curvature. The convergence of interfaces as c -+ 0 is in the Hausdorff distance sense

Domain dependent monotonicity formula for a singular perturbation problem

We consider a singular perturbation problem arising in the scalar phase field model with Keumann boundary conditions on convex domains, and establish a monotonicity formula for general critical points. This gives the Hausdorff distance convergence of the phase boundaries as the parameters tend to ,1ero. We apply the result to stable critical points defined on strictly convex domains, showing that the limit interfaces for stable critical points arc necessarily connected

Remarks on convergence of the Allen-Cahn equation

We answer a question posed by Ilmanen on the integrality of varifolds which appear as the singular perturbation limit of the Allen-Cahn equation. We show that the density of the limit measure is integer multiple of the surface constant 1-1,n-1-a.e. for a.e. time. This shows that limit measures obtained via the Allen­Cahn equation and those via Brakke's construction share the same integrality property as well as being weak solutions for the mean curvature flow equation

On stable critical points for a singular perturbation problem

We consider a singular perturbation problem arising in the scalar phase field model which [-converges to the area func­ tional. Assuming the stability of the critical points for s-problcms, we show that the interface regions converge to a generalized stable minimal hypcrsurfacc as s --'t 0. The limit has L2 generalized second fundamental form and the stability condition is expressed in terms of the corresponding inequality satisfied by stable minimal hypcrsurfaccs. We show that the limit is a finite number of lines with no intersections when the dimension of the domain is 2