### A new method for numerical solution of checkerboard fields.

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We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies$$\phantom{\rule{-17.07164pt}{0ex}}{\mathcal{E}}_{\epsilon}\left(m\right)={\int}_{\Omega}\phi \left(x,\frac{x}{\epsilon},m\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x-{\int}_{\Omega}{h}_{e}\left(x\right)\xb7m\left(x\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+\frac{1}{2}{\int}_{{\mathbb{R}}^{3}}{\left|\nabla u\left(x\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$of a large ferromagnetic body is obtained.

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $${\mathcal{E}}_{\epsilon}\left(m\right)={\int}_{\Omega}\phi \left(x,\frac{x}{\epsilon},m\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x-{\int}_{\Omega}{h}_{e}\left(x\right)\xb7m\left(x\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+\frac{1}{2}{\int}_{{\mathbb{R}}^{3}}{\left|\nabla u\left(x\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$ of a large ferromagnetic body is obtained.