The Nonlinear Sieve Problem and Applications to Thin Films

We consider variational problems defined on domains ‘weakly’ connected through a separation hyperplane (‘sieve plane’) by an ε-periodically distributed ‘contact zone’. We study the asymptotic behaviour as ε tends to 0 of integral functionals in such domains in the nonlinear and vector-valued case, showing that the asymptotic memory of the sieve is described by a nonlinear ‘capacitarytype’ formula. In particular we treat the case when the integral energies on both sides of the sieve plane satisfy different growth conditions. We also study the case of thin films, with flat profile and thickness ε, connected by the same sieve plane.