## HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, H-MEASURES, AND DISCRETE WIGNER MEASURES

Citations: | 1 - 0 self |

### BibTeX

@MISC{Firoozye_homogenizationon,

author = {Nikan B. Firoozye},

title = {HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, H-MEASURES, AND DISCRETE WIGNER MEASURES},

year = {}

}

### OpenURL

### Abstract

Abstract. We fully characterize the small-parameter 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 small-parameter limit, we use discrete Wigner transforms to transform the stored-energy 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 small-parameter 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 H-measure limit, and the limiting magnetization. We show that the classical

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Citation Context ...j jh j 2 jj L 1 (Q) = jj ~ d jj 2 L 2 (< N ) C 2 and that, due to the homogeneity and cancellation property, K 2 L1 (Q). In fact we can say more about K following Wainger, ([22], see also Stein-Weiss =-=[18]-=-, ch. 7, x6.1): Theorem 1 (Wainger,1965). Let K(x) =jxj K0 (x=jxj) for N, with K 0 2 C 1 (S N 1 ) and Z SN 1 and, without loss of generality, assume that K 0 (s) = K 0 (s)ds =0; 1X X l=1 m al;mYl;m(s)... |

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Citation Context ... transform of the sequence d and has the property that U 2 l 1 (L =2) L 1 (T N ). It also satis es the property X r2L =2 U (r; )=jd j 2 ( ); if ~ d 2 l1 (L ), where d ( )= P Ld( n)e in , (see Folland =-=[6]-=- for other properties of the continuous Wigner transform). Markowich recently used the Wigner transform for converting Schrodinger equations into Vlasov-Liouville equations [16]. Gerard [9] and Lions-... |

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Citation Context ... extensively in the study of mechanics in elds ranging from classical magnetostatics to the quantum mechanical structures of materials, (e.g., Bloch [2]), to the modeling of coherent structures (Toda =-=[20]-=-, among others). Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius [21], Fujiki et al [7], De'Bell and Whitehead [4... |

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Citation Context ... studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius [21], Fujiki et al [7], De'Bell and Whitehead [4], James-Muller [10], Khachaturyan =-=[13]-=-, etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized for every value of the lattice parameter, or in the more general context, where com... |

37 |
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Citation Context ...essential lters to separate various lengthscales, we are able to characterize the limiting energy completely by three complementary tools: the limiting magnetization; the H-measure limit, (see Tartar =-=[19]-=-, Gerard [8]), to represent oscillations on a much larger lengthscale than that of the lattice; and the discrete Wigner measure limit, used to represent oscillations only on the scale of the lattice. ... |

9 |
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Citation Context ...( n)e in , (see Folland [6] for other properties of the continuous Wigner transform). Markowich recently used the Wigner transform for converting Schrodinger equations into Vlasov-Liouville equations =-=[16]-=-. Gerard [9] and Lions-Paul [15] have expanded upon this use, introducing the corresponding continuous Wigner measure solutions to Vlasov-Liouville equations as a tool for obtaining semiclassical limi... |

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Citation Context ...ying upon the work of S. Wainger [22] on the Fourier transforms of kernels on lattices. In the process, we introduce discrete Wigner measures, the continuous analogs of which have been used by Gerard =-=[9]-=- and Lions-Paul [15] to describe semiclassical limits of Schrodinger equations. After transforming the problem to one on the torus, we cannot pass to the limit directly; it is rst necessary to separat... |

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Citation Context ...o the modeling of coherent structures (Toda [20], among others). Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius =-=[21]-=-, Fujiki et al [7], De'Bell and Whitehead [4], James-Muller [10], Khachaturyan [13], etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized ... |

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Citation Context ...and small. Thus, the right hand side of (26) tends to zero. Continuing, we nowturn to the justi cation of (25b). We see that K ( ) ! K( ) pointwise on Q, and thus K * K weak-? in Cper(Q), (see Kaplan =-=[12]-=-, x54.2), and also uniformly on Q n B for >0. Using this, we estimate the following: Z Z Z Z K( )(1 ( = ))d ( ) K d (K K )d + K ( = )d Z 17 Q Q (K K )d + jjKjj L 1 (d ) Z ( = )d :We rst claim that jj... |

2 |
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Citation Context ...oherent structures (Toda [20], among others). Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius [21], Fujiki et al =-=[7]-=-, De'Bell and Whitehead [4], James-Muller [10], Khachaturyan [13], etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized for every value of... |

1 |
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Citation Context ...ctrical networks, (rather than lattices), but only under the in uence of speci c short-range \exchange energies." Without considering small-parameter limits, Khachaturyan [14] and Chouliourous-Pouget =-=[3]-=- also used lattice models to describe the evolution of magnetic moments with more complicated Hamiltonians (higher-order interactions). Using nite Fourier transforms and theta-functions as a means of ... |

1 |
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(Show Context)
Citation Context ...0], among others). Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius [21], Fujiki et al [7], De'Bell and Whitehead =-=[4]-=-, James-Muller [10], Khachaturyan [13], etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized for every value of the lattice parameter, or ... |

1 |
Firoozye and V.Sverak, Measure lters: extensions of Wiener's theorem for measures and BV functions, (in preparation
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(Show Context)
Citation Context ...iting energies. In sections 6 and 7, we describe the unusual behavior of limiting energies of weakshort oscillations. In section 6, we use the theory of measure lters developed by Firoozye and Sverak =-=[5]-=- to give an example of a sequence which oscillates on the scale of the lattice and to describe the limiting energy of a localization of this sequence to a given domain. We show that the limiting energ... |

1 |
Compacite par compensation et regularite deux-microlocale
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Citation Context ...rs to separate various lengthscales, we are able to characterize the limiting energy completely by three complementary tools: the limiting magnetization; the H-measure limit, (see Tartar [19], Gerard =-=[8]-=-), to represent oscillations on a much larger lengthscale than that of the lattice; and the discrete Wigner measure limit, used to represent oscillations only on the scale of the lattice. 2for The \n... |

1 |
Internal variables and ne-scale oscillations in micromagnetics
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Citation Context ... Lattice models are studied especially in their relation to continuum models through the use of small-parameter limits, (e.g., Vogelius [21], Fujiki et al [7], De'Bell and Whitehead [4], James-Muller =-=[10]-=-, Khachaturyan [13], etc). Smallparameter limits have been studied in the context of -convergence, where some storedenergy is minimized for every value of the lattice parameter, or in the more general... |