A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functionalanalytic approach

In this paper, we consider two bounded open subsets of $\Omega^{i}$ and $\Omega^{o}$ of ${\mathbb{R}}^{n}$ containing $0$ and a (nonlinear) function $G^{o}$ of $\partial\Omega^{o}\times{\mathbb{R}}^{n}$ to ${\mathbb{R}}^{n}$, and a map $T$ of $]1-(2/n),+\infty[$ times the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries to $M_{n}({\mathbb{R}})$, and we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0 & {\mathrm{in}}\ \Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}\,, \\ -T(\omega,Du)\nu_{\epsilon\Omega^{i}}=0 & {\mathrm{on}}\ \epsilon\partial\Omega^{i}\,, \\ T(\omega,Du(x))\nu^{o}(x)=G^{o}(x,u(x)) & \forall x\in\partial \Omega^{o}\,, \end{array} \right. \] where $\nu_{\epsilon\Omega^{i}}$ and $\nu^{o}$ denote the outward unit normal to $\epsilon\partial \Omega^{i}$ and $\partial\Omega^{o}$, respectively, and where $\epsilon>0$ is a small parameter. Here $(\omega-1)$ plays the role of ratio between the first and second Lam\'{e} constants, and $T(\omega,\cdot)$ plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and $G^{o}$ plays the role of (a constant multiple of) a traction applied on the points of $\partial\Omega^{o}$. Then we prove that under suitable assumptions the above problem has a family of solutions $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ for $\epsilon'$ sufficiently small and we show that in a certain sense $\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}$ can be continued real analytically for negative values of $\epsilon$.