A singularly perturbed nonlinear traction problem in a periodically perforated domain: A functional analytic approach

We consider a periodically perforated domain obtained by making in $\mathbb{R}^n$ a periodic set of holes, each of them of size proportional to $\epsilon$. Then we introduce a nonlinear boundary value problem for the Lam\'e equations in such a periodically perforated domain. The unknown of the problem is a vector valued function $u$ which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size $\epsilon$ contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then our aim is to describe what happens to the displacement vector function $u$ when $\epsilon$ tends to $0$. Under suitable assumptions we prove the existence of a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon'[}$ with a prescribed limiting behaviour when $\epsilon$ approaches $0$. Moreover, the family $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon'[}$ is in a sense locally unique and can be continued real analytically for negative values of $\epsilon$.