On Full Two-Scale Expansion of the Solutions of Nonlinear Periodic Rapidly Oscillating Problems and Higher-Order Homogenised Variational Problems.

We consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations, which may be a model of, e.g., nonlinear conducting, dielectric, or deforming in a restricted way hardening elastic-plastic composites, with “outer” periodicity conditions of a fixed large period. Under some natural growth assumptions on the stored-energy function, we construct for uniformly elliptic problems a full two-scale asymptotic expansion, which has a precise “double-series” structure, separating the slow and the fast variables “in all orders”, so that its “slowly varying” part solves asymptotically an “infinite-order homogenised equation” (cf. Bakhvalov, N.S., Panasenko, G.P.:Homogenisation: Averaging Processes in Periodic Media. Nauka, Moscow, 1984 (in Russian); English translation: Kluwer, 1989), and whose higher-order terms depend on the higher gradients of the slowly varying part. We prove the error bound, i.e., that the truncated asymptotic expansion is “higher-order” close to the actual solution in appropriate norms. The approach is extended to a non-uniformly elliptic case: for two-dimensional power-law potentials we prove the “non-degeneracy” using topological index methods. Examples and explicit formulae for the higher-order terms are given. In particular, we prove that the first term in the higher-order homogenised equations is related to the first-order corrector to the “mean” flux, and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest (second) derivative being a linear combination of the second minors of the Hessian with coefficients depending on the first gradient, and in dimension two is of Monge-Ampère type. We show that this term is present at least for some examples (three-phase power-law laminates).In the second part of the paper we extend to this nonlinear context some of the results previously developed by us in the linear case (Smyshlyaev, V.P., Cherednichenko, K.D.J. Mech. Phys. Solids48, 1325-1357, 2000). In particular, we prove that the slowly varying part of the full asymptotic expansion is the rigorous asymptotics “in all orders” for the “translationally averaged” actual solution and flux, or in the sense of a higher-order version of the weak convergence. We then explore to what extent the method of variational truncation of the infinite-order homogenised equation, successfully implemented by us in the linear context in the previous work for constructing explicit “higher-order” homogenised equations, is extendable to the nonlinear regime. We propose a natural extension and prove that at least under some further natural “non-degeneracy” assumptions it has a solution (the existence), and that any such solution is close to the actual solution in appropriate norms. [ABSTRACT FROM AUTHOR]