Bounds on the effective behavior of a homogenized generalized Reynolds equation (Q2471579)

Summary: We study upper and lower bounds for estimating the effective behavior described by homogenizing a problem of the type \[ {\partial\over\partial x_1}\Biggl(a_1\Biggl(x, {x\over\varepsilon}\Biggr) {\partial u_\varepsilon\over\partial x_1}- b_1\Biggl(x,{x\over \varepsilon}\Biggr)\Biggr)+ {\partial\over\partial x_2}\Biggl(a_2\Biggl(x,{x\over\varepsilon}\Biggr) {\partial u_\varepsilon\over\partial x_2}- b_2\Biggl(x, {x\over\varepsilon}\Biggr)\Biggr)= f(x), \] which is a generalization of the Reynold equation. All cases when these bounds coincide are also found.