Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach

Abstract

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 & {\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  $T(\omega,\cdot)$ denotes (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and $\delta_{2,n}$ denotes the Kronecker symbol.  Under the condition that $\gamma$ generates a very strong singularity, \textit{i.e.}, the  case in which  $\lim_{\epsilon\to 0^{+}}\frac{\gamma(\epsilon)}{\epsilon^{n-1}}$  exists in $ [0,+\infty[$, 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 behavior of such a family as $\epsilon$  is close to $0$ by an approach which is alternative to those of asymptotic analysis.