Scale-transformations and homogenization of maximal monotone relations with applications

Abstract

In homogenization, two-scale models arise, e.g., by applying Nguetseng's notion of two-scale convergence to nonlinear PDEs. A homogenized single-scale problem may then be derived via scale-transformations. A variational formulation due to Fitzpatrick is here used for the scale-integration of two-scale maximal monotone relations, and for the converse operation of scale-disintegration. These results are applied to the periodic homogenization of a quasilinear model of Ohmic electric conduction with Hall effect:

$\lefteqn{\vec{E}\in\vec{\alpha}(\vec{J},x/\varepsilon )+h(x/\varepsilon )\vec{J}\times\vec{B}(x/\varepsilon )+\vec{E}_{a}(x/\varepsilon ),}$

$\lefteqn{\nabla\times\vec{E}=\vec{g}(x/\varepsilon ),\qquad\nabla\cdot\vec{J}=0\quad\mbox{in }\varOmega,}$

with $\vec{\alpha}(\cdot,x/\varepsilon )$ maximal monotone, $\vec{B},\vec{E}_{a},h,\vec{g}$ prescribed fields. (This corresponds to a quasilinear second-order elliptic equation in curl form: $\nabla\times\vec{\beta}(\nabla\times u,x/\varepsilon )=\vec{g}(x/\varepsilon )$ .) This result is also retrieved via De Giorgi's Γ-convergence.