Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

Let $\Omega^{i}$ and $\Omega^{o}$ be two bounded open subsets of ${\mathbb{R}}^{n}$ containing $0$. Let $G^{i}$ be a (nonlinear) map from $\partial\Omega^{i}\times {\mathbb{R}}^{n}$ to $ {\mathbb{R}}^{n}$. Let $a^{o}$ be a map from $\partial\Omega^{o}$ to the set $M_{n}({\mathbb{R}})$ of $n\times n$ matrices with real entries. Let $g$ be a function from $\partial\Omega^{o}$ to ${\mathbb{R}}^{n}$. Let $\gamma$ be a positive valued function defined on a right neighborhood of $0$ in the real line. Let $T$ be a map from $]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})$ to $M_{n}({\mathbb{R}})$. Then we consider the problem \[ \left\{ \begin{array}{ll} {\mathrm{div}}\, (T(\omega,Du))=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad \quad \qquad {\mathrm{in}} &\Omega^{o}\setminus\epsilon{\mathrm{cl}}\Omega^{i}, \\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}(x/ \epsilon, \gamma(\epsilon)\epsilon^{-1} (\log \epsilon)^{-\delta_{2,n}} u(x)) & \forall x\in \epsilon\partial\Omega^{i}, \\ T(\omega,Du(x))\nu^{o}(x)=a^{o}(x)u(x)+g(x) & \forall x\in\partial \Omega^{o}, \end{array} \right. \] 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. Under the condition that $\lim_{\epsilon\to 0^{+}}{\gamma(\epsilon)}{\epsilon^{-1}(\log\epsilon)^{-\delta_{2,n}}}=0$ and $\lim_{\epsilon\to 0^{+}}{\epsilon^{n-1}}{\gamma(\epsilon)^{-1}}=0$, 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 analyze the behaviour of such a family as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. Here $\delta_{2,n}$ denotes the Kronecker symbol.