A note on bounds for non-linear multivalued homogenized operators (Q1978989)

The paper deals with the homogenization problem for equations with nonlinear maximal monotone multivalued operators of the type \[ - \text{div}\left (a\left ({x\over \varepsilon}, Du_\varepsilon \right)\right) \ni f \quad \text{in } \Omega, \] that appear in nonlinear elasticity and plasticity. Since the computation of homogenized operators is difficult, their bounds and estimates are useful. The author generalizes estimates of Reuss-Voigt-Wiener and Hahin-Shtrikman type to nonlinear monotone operators of potential type mentioned above. The proof is based on a variational principle using means of convex analysis, namely duality mapping and Yosida approximation.