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HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence ..."
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
DIRECTIONAL OSCILLATIONS, CONCENTRATIONS, AND COMPENSATED COMPACTNESS VIA MICROLOCAL COMPACTNESS FORMS
, 2014
"... ABSTRACT. This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in Lpbounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending ..."
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ABSTRACT. This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in Lpbounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and Hmeasures. In Lpspaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow one to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily— a distinct advantage over Young measure theory. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar’s framework of compensated compactness; consequently, we establish a new weaktostrong compactness theorem in a “geometric ” way. After developing several aspects of the abstract theory, we consider three applications: For lamination microstructures, the hierarchy of oscillations is reflected in the MCF. The directional information retained in an MCF is harnessed in the relaxation theory for anisotropic integral functionals. Finally, we indicate how the theory pertains to the study of propagation of singularities in certain systems of PDEs. The proofs combine measure theory, Young measures, and harmonic analysis.