\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 242, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/242\hfil Local uniqueness] {Local uniqueness for singularly perturbed periodic nonlinear traction problems} \author[M. Dalla Riva, P. Musolino \hfil EJDE-2014/242\hfilneg] {Matteo Dalla Riva, Paolo Musolino} % in alphabetical order \address{Matteo Dalla Riva \newline Centro de Investiga\c{c}\~ao e Desenvolvimento em Matem\'atica e Aplica\c{c}\~oes (CIDMA), Universidade de Aveiro, Portugal} \email{matteo.dallariva@gmail.com} \address{Paolo Musolino \newline Dipartimento di Matematica, Universit\`a degli Studi di Padova, Italy} \email{musolinopaolo@gmail.com} \thanks{Submitted March 21, 2014. Published November 18, 2014.} \subjclass[2000]{35J65, 31B10, 45F15, 74B05} \keywords{Nonlinear traction problem; singularly perturbed domain; \hfill\break\indent linearized elastostatics; local uniqueness; integral representation; elliptic system} \begin{abstract} We present a limiting property and a local uniqueness result for converging families of solutions of a singularly perturbed nonlinear traction problem in an unbounded periodic domain with small holes. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we use an argument based on functional analysis and potential theory to show a limiting property and a local uniqueness result for families of solutions of a singularly perturbed nonlinear traction problem in linearized elasticity. We fix once for all \[ n\in {\mathbb{N}}\setminus\{0,1 \}\,, \quad q_{11},\dots,q_{nn}\in]0,+\infty[\, , \quad Q\equiv\Pi_{j=1}^{n}]0,q_{jj}[\, . \] Then we denote by $q$ the $n\times n$ diagonal matrix with diagonal entries $q_{11}, \dots, q_{nn}$. We also assume that \[ \parbox{10cm}{ $\alpha\in]0,1[$ and $\Omega^h \subseteq {\mathbb{R}}^{n}$ is bounded, open, connected, of class $C^{1,\alpha}$, containing the origin $0$, and with a connected exterior ${\mathbb{R}}^{n}\setminus\operatorname{cl}\Omega^h$.} \] Here $\operatorname{cl}$ denotes the closure and the letter `$h$' stands for `hole'. The set $\Omega^{h}$ will play the role of the shape of the perforation. Moreover, we fix \[ \text{$p\in Q$ and $\epsilon_{0}\in]0,+\infty[$ such that $p+\epsilon \operatorname{cl}\Omega^h\subseteq Q$ for all $\epsilon\in]-\epsilon_{0},\epsilon_{0}[$}\,. \] To shorten our notation, we set \[ \Omega^h_{p,\epsilon} \equiv p+\epsilon\Omega^h \] and we define the periodically perforated domain \[ {\mathbb{S}} [\Omega^h_{p,\epsilon}]^{-}\equiv {\mathbb{R}}^{n}\setminus\cup_{z \in \mathbb{Z}^n} \operatorname{cl}(\Omega^h_{p,\epsilon}+qz ) \] for all $\epsilon\in ]-\epsilon_{0},\epsilon_{0}[ $. A function $u$ defined on $\operatorname{cl}{\mathbb{S}}[\Omega_{p,\epsilon}^{h}]^{-}$ is said to be $q$-periodic if \[ u(x+q z)=u(x)\quad\forall x\in \operatorname{cl}{\mathbb{S}}[\Omega_{p,\epsilon}^{h}]^{-}\,, \quad \forall z \in \mathbb{Z}^n\, . \] We now introduce a nonlinear traction boundary value problem in ${\mathbb{S}} [\Omega^h_{p,\epsilon}]^{-}$. To do so, we denote by $T$ the function from $ ]1-(2/n),+\infty[\times M_n(\mathbb{R})$ to $M_n(\mathbb{R})$ which takes the pair $(\omega,A)$ to \[ T(\omega,A)\equiv (\omega-1)(\operatorname{tr}A)I_n+(A+A^t) \,. \] Here $M_n(\mathbb{R})$ denotes the space of $n\times n$ matrices with real entries, $I_n$ denotes the $n\times n$ identity matrix, $\operatorname{tr}A$ and $A^t$ denote the trace and the transpose matrix of $A$, respectively. We observe that $(\omega-1)$ plays the role of the ratio between the first and second Lam\'e constants and that the classical linearization of the Piola Kirchoff tensor equals the second Lam\'e constant times $T(\omega,\cdot)$ (cf., e. g., Kupradze et al \cite{KuGeBaBu79}). We also note that \[ \operatorname{div} T(\omega, Du )=\Delta u+\omega\nabla \operatorname{div} u\, , \] for all regular vector valued functions $u$. Now let $G$ be a function from $\partial \Omega^h \times\mathbb{R}^n$ to $\mathbb{R}^n$, let $B \in M_n(\mathbb{R})$, and let $\epsilon \in ]0,\epsilon_0[$. We introduce the nonlinear traction problem \begin{equation}\label{bvp:nltraceleps} \begin{gathered} \operatorname{div} T(\omega, Du )= 0 \quad \text{in } \mathbb{S} [\Omega^h_{p,\epsilon}]^-\,, \\ u(x+qe_j) =u(x) +Be_j \quad \forall x \in \operatorname{cl} {\mathbb{S}} [\Omega^h_{p,\epsilon}]^{-}, \forall j\in \{1,\dots,n\}, \\ T(\omega,Du(x))\nu_{\Omega^h_{p,\epsilon}}(x)=G\bigl((x-p)/\epsilon,u(x)\bigr) \quad \forall x \in \partial \Omega^h_{p,\epsilon}\,, \end{gathered} \end{equation} where $\nu_{\Omega^h_{p,\epsilon}}$ denotes the outward unit normal to $\partial \Omega^h_{p,\epsilon}$ and $\{e_1,\dots, e_n\}$ denotes the canonical basis of $\mathbb{R}^n$. Because of the presence of a nonlinear term in the third equation of problem \eqref{bvp:nltraceleps}, we cannot claim in general the existence of a solution. However, we know by \cite{DaMu14} that under suitable assumptions there exists $\epsilon_1\in]0,\epsilon_0]$ such that the boundary value problem in \eqref{bvp:nltraceleps} has a solution $u(\epsilon,\cdot)$ in $C^{1,\alpha}_{\mathrm{loc}}(\operatorname{cl}\mathbb{S}[{\Omega^h_{p,\epsilon}}]^-, \mathbb{R}^n)$ for all $\epsilon\in]0,\epsilon_1[$. Moreover, the family $\{u(\epsilon,\cdot)\}_{\epsilon\in]0,\epsilon_1[}$ is uniquely determined (for $\epsilon$ small) by its limiting behavior as $\epsilon$ tends to $0$ and the dependence of $u(\epsilon,\cdot)$ upon the parameter $\epsilon$ can be described in terms of real analytic maps of $\epsilon$ defined in an open neighborhood of $0$. In this article, we study the limiting behavior and the local uniqueness of families of solutions of problem \eqref{bvp:nltraceleps}, under weaker assumptions than those in \cite{DaMu14}. In particular, in Theorem \ref{thm:rem}, we show that if $\{\varepsilon_j\}_{j \in \mathbb{N}}$ is a sequence in $]0,\epsilon_0[$ converging to $0$ and if $\{u_j\}_{j \in \mathbb{N}}$ is a family of functions such that $u_j$ solves problem \eqref{bvp:nltraceleps} for $\epsilon = \varepsilon_j$ and such that the restrictions to $\partial\Omega^h$ of the rescaled functions $u_j(p+\varepsilon_j\cdot)$ converge to a function $v_\ast$ as $j$ tends to $+\infty$, then $v_\ast$ must be equal to a constant vector $\xi_\ast \in \mathbb{R}^n$ and $u_j$ converges to $\xi_\ast + Bq^{-1}(\cdot-p)$ uniformly on bounded open subsets of $\mathbb{R}^n \setminus (p+q\mathbb{Z}^n)$. In Theorem \ref{localuniq}, instead, we prove that, under suitable assumptions, if $\{\varepsilon_j\}_{j \in \mathbb{N}}$ is a sequence in $]0,\epsilon_0[$ converging to $0$ and if $\{u_j\}_{j \in \mathbb{N}}$, $\{v_j\}_{j \in \mathbb{N}}$ are families of functions such that $u_j$ and $v_j$ solve problem \eqref{bvp:nltraceleps} for $\epsilon = \varepsilon_j$ and such that the restrictions to $\partial\Omega^h$ of $u_j(p+\varepsilon_j\cdot)$ and of $v_j(p+\varepsilon_j\cdot)$ converge to the same function, then we must have $u_j=v_j$ for $j$ big enough. We also note that the present article extends to the case of a nonlinear traction problem the results of \cite{DaLaMu13}, concerning a nonlinear Robin problem for the Laplace equation. The functional analytic approach adopted in \cite{DaMu14} and in the present paper for the investigation of the behavior of the solutions of problem \eqref{bvp:nltraceleps} has been previously exploited by Lanza de Cristoforis and the authors to analyze singular perturbation problems for the Laplace operator in \cite{DaMu12, La07}, for the Lam\'e equations in \cite{DaLa10a, DaLa10b, DaLa11}, and for the Stokes system in \cite{Da13}. Concerning problems in an infinite periodically perforated domain, we mention in particular \cite{DaMu13, DaMu14, LaMu13, Mu12}. We note that singularly perturbed boundary value problems have been largely investigated with the methods of asymptotic analysis. As an example, we mention the works of Beretta et al \cite{BeBoFrMa12}, Bonnaillie-No\"el et al \cite{BoDaToVi09}, Iguernane et al \cite{IgNaRoSoSz09}, Maz'ya et al \cite{MaMoNi13}, Maz'ya et al \cite{MaNaPl00}, Nazarov et al \cite{NaRuTa12}, Nazarov and Sokolowski \cite{NaSo03}, and Vogelius and Volkov \cite{VoVo00}. In particular, in connection with periodic problems, we mention, e. g., Ammari et al \cite{AmKaLi06}. Moreover, for problems in periodic domains, we mention the method of functional equations and, for example, the works of Castro et al \cite{CaPeRo09} and Drygas and Mityushev \cite{DrMi09} This article is organized as follows. Section \ref{not} is a section of notation and preliminaries. In Section \ref{intform} we provide an integral formulation of problem \eqref{bvp:nltraceleps}. In Section \ref{conv} we prove our main results on the limiting behavior and the local uniqueness of a family of solutions of problem \eqref{bvp:nltraceleps}. \section{Notation and preliminaries}\label{not} Let $\mathcal{X}$ and $\mathcal{Y}$ be normed spaces. We denote by $\mathcal{L}(\mathcal{X},\mathcal{Y})$ the space of linear and continuous maps from $\mathcal{X}$ to $\mathcal{Y}$, equipped with its usual norm of the uniform convergence on the unit sphere of $\mathcal{X}$. We denote by $I$ the identity operator. The inverse function of an invertible function $f$ is denoted $f^{(-1)}$, as opposed to the reciprocal of a real-valued function $g$, or the inverse of a matrix $B$, which are denoted $g^{-1}$ and $B^{-1}$, respectively. If $B$ is a matrix, then $B_{ij}$ denotes the $(i,j)$ entry of $B$. If $x\in\mathbb{R}^n$, then $x_{j}$ denotes the $j$-th coordinate of $x$ and $|x|$ denotes the Euclidean modulus of $ x$. A dot `$\cdot$' denotes the inner product in ${\mathbb R}^{n}$. For all $R>0$ and all $x\in{\mathbb{R}}^{n}$ we denote by ${\mathbb{B}}_{n}( x,R)$ the ball $\{y\in{\mathbb{R}}^{n}: | x- y|2\,, \end{cases} \] where $s_{n}$ denotes the $(n-1)$-dimensional measure of $\partial{\mathbb{B}}_{n}(0,1)$. $S_{n}$ is well-known to be the fundamental solution of the Laplace operator. Let $\omega \in ]1-(2/n),+\infty[$. We denote by $\Gamma_{n,\omega}$ the matrix valued function from $\mathbb{R}^n \setminus \{0\}$ to $M_{n}(\mathbb{R})$ which takes $x$ to the matrix $\Gamma_{n,\omega}(x)$ with $(j,k)$ entry defined by \[ \Gamma_{n,\omega,j}^k(x)\equiv \frac{\omega+2}{2(\omega+1)}\delta_{j,k}S_n(x) -\frac{\omega}{2(\omega+1)}\frac{1}{s_n}\frac{x_j x_k}{|x|^n} \quad\forall (j,k)\in\{1,\dots,n\}^2\,, \] where $\delta_{j,k}=1$ if $j=k$, $\delta_{j,k}=0$ if $j \neq k$. As is well known, $\Gamma_{n,\omega}$ is the fundamental solution of the operator $L[\omega] \equiv \Delta+\omega \nabla \operatorname{div}$. We find also convenient to set \[ \Gamma_{n,\omega}^k\equiv \bigl(\Gamma_{n,\omega,j}^k\bigr)_{j \in \{1,\dots,n\}}\,, \] which we think as a column vector for all $k\in\{1,\dots,n\}$. Now let $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. Then we set \[ v[\omega,\mu](x)\equiv \int_{\partial \Omega}\Gamma_{n,\omega}(x-y)\mu(y) \,d\sigma_y\,, \] for all $x \in \mathbb{R}^n$ and for all $\mu \equiv (\mu_j)_{j\in \{1,\dots,n\}} \in C^{0,\alpha} (\partial \Omega,\mathbb{R}^n)$. Here $d\sigma$ denotes the area element on $\partial\Omega$. As is well known, the elastic single layer potential $v[\omega,\mu]$ is continuous in the whole of $\mathbb{R}^n$. We set $v^+[\omega,\mu]\equiv v[\omega,\mu]\big|_{\operatorname{cl} \Omega}$ and $v^-[\omega,\mu]\equiv v[\omega,\mu]\big|_{\mathbb{R}^n \setminus \Omega}$. We also find convenient to set \[ w_{\ast}[\omega, \mu](x)\equiv \int_{\partial {\Omega}}\sum_{l=1}^n \mu_{l}(y) T(\omega,D\Gamma_{n,\omega}^{l}(x-y))\nu_{{\Omega}}(x)\,d\sigma_y \quad \forall x \in \partial {\Omega}\,. \] Here $\nu_{\Omega}$ denotes the outward unit normal to $\partial\Omega$. For properties of elastic layer potentials, we refer, e. g., to \cite[Appendix A]{DaLa10a}. We now introduce a periodic analogue of the fundamental solution of $L[\omega]$ (cf., e. g., Ammari et al \cite[Lemma 3.2]{AmKaLi06}, \cite[Thm.~3.1]{DaMu14}). Let $\omega \in ]1-(2/n),+\infty[$. We denote by $\Gamma_{n,\omega}^{q}\equiv (\Gamma_{n,\omega,j}^{q,k})_{(j,k)\in\{1,\dots,n\}^2}$ the matrix of distributions with $(j,k)$ entry defined by \[ \Gamma_{n,\omega,j}^{q,k}(x)\equiv \sum_{z \in \mathbb{Z}^n \setminus \{0\}} \frac{1}{4 \pi^2 |Q| |q^{-1}z|^2} \Big[ -\delta_{j,k}+\frac{\omega}{\omega+1}\frac{(q^{-1}z)_j (q^{-1}z)_k}{|q^{-1}z|^2}\Big]e^{2\pi i (q^{-1} z)\cdot x} \] for all $(j,k) \in \{1,\dots,n\}^2$, where the series converges in the sense of distributions. Then \[ L[\omega] \Gamma_{n,\omega}^{q} =\sum_{z \in \mathbb{Z}^n}\delta_{qz}I_n-\frac{1}{ |Q|}I_n \,, \] where $\delta_{qz}$ denotes the Dirac measure with mass at $qz$ for all $z \in \mathbb{Z}^n$. Moreover, $\Gamma_{n,\omega}^{q}$ is real analytic from $\mathbb{R}^n \setminus q\mathbb{Z}^n$ to $M_n(\mathbb{R})$ and the difference $\Gamma_{n,\omega}^{q}-\Gamma_{n,\omega}$ can be extended to a real analytic function from $(\mathbb{R}^n \setminus q \mathbb{Z}^n) \cup\{0\}$ to $M_n(\mathbb{R})$ which we denote by $R^q_{n,\omega}$. We find convenient to set \[ \Gamma_{n,\omega}^{q,k}\equiv \bigl(\Gamma_{n,\omega,j}^{q,k}\bigr)_{j \in \{1,\dots, n\}}\,,\quad R_{n,\omega}^{q,k}\equiv \bigl(R_{n,\omega,j}^{q,k}\bigr)_{j \in \{1,\dots,n\}}\,, \] which we think as column vectors for all $k\in\{1,\dots,n\}$. Let ${\Omega_Q}$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$ such that $\operatorname{cl}{\Omega_Q}\subseteq Q$. Let $\mu \in C^{0,\alpha}(\partial {\Omega_Q},\mathbb{R}^n)$. Then we denote by $v_q[\omega, \mu]$ the periodic single layer potential, namely the $q$-periodic function from $\mathbb{R}^n$ to $\mathbb{R}^n$ defined by \[ v_q[\omega, \mu](x)\equiv \int_{\partial {\Omega_Q}}\Gamma^q_{n,\omega}(x-y) \mu(y)\,d\sigma_y \quad \forall x \in \mathbb{R}^n\,. \] We also find convenient to set \[ w_{q,\ast}[\omega, \mu](x)\equiv \int_{\partial {\Omega_Q}}\sum_{l=1}^n \mu_{l}(y) T(\omega,D\Gamma_{n,\omega}^{q,l}(x-y))\nu_{{\Omega_Q}}(x)\,d\sigma_y \quad \forall x \in \partial {\Omega_Q}\,. \] Here $\nu_{\Omega_Q}$ denotes the outward unit normal to $\partial\Omega_Q$. If $\mu\in C^{0,\alpha}(\partial{\Omega_Q},\mathbb{R}^n)$, then the function $v^{+}_{q}[\omega,\mu]\equiv v_{q}[\omega,\mu]\big|_{\operatorname{cl}{\mathbb{S}} [{\Omega_Q}]}$ belongs to $C^{1,\alpha}_{\mathrm{loc}}(\operatorname{cl}{\mathbb{S}} [{\Omega_Q}],\mathbb{R}^n)$ and the function $v^{-}_{q}[\omega,\mu]\equiv v_{q}[\omega,\mu]\big|_{\operatorname{cl}{\mathbb{S}} [{\Omega_Q}]^{-}}$ belongs to $C^{1,\alpha}_{\mathrm{loc}} (\operatorname{cl}{\mathbb{S}}[{\Omega_Q}]^{-},\mathbb{R}^n)$. For further properties of $v_q[\omega,\cdot]$ and $w_{q,\ast}[\omega,\cdot]$ we refer the reader to \cite[Thm.~3.2]{DaMu14}. \section{An integral equation formulation of the nonlinear traction problem} \label{intform} In this section we provide an integral formulation of problem \eqref{bvp:nltraceleps} (cf.~\cite[\S 5]{DaMu14}). We use the following notation. If $G\in C^0(\partial \Omega^h \times \mathbb{R}^n,\mathbb{R}^n)$, then we denote by $F_G$ the (nonlinear nonautonomous) composition operator from $C^0(\partial \Omega^h,\mathbb{R}^n)$ to itself which takes $v \in C^0(\partial \Omega^h,\mathbb{R}^n)$ to the function $F_G[v]$ from $\partial \Omega^h$ to $\mathbb{R}^n$ defined by \[ F_G[v](t)\equiv G(t,v(t)) \quad \forall t \in \partial \Omega^h\, . \] Then we consider the following assumptions \begin{equation}\label{assG} G \in C^0(\partial \Omega^h \times \mathbb{R}^n,\mathbb{R}^n)\, ,\quad \text{$F_G$ maps $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)$ to itself.} \end{equation} We also note here that if $F_G$ is continuosly Fr\'echet differentiable from $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)$ to itself, then the gradient matrix $D_uG(\cdot,\cdot)$ of $G(\cdot,\cdot)$ with respect to the variable in $\mathbb{R}^n$ exists. Moreover, $D_uG(\cdot,\xi)\in C^{0,\alpha}(\partial \Omega^h,M_{n}(\mathbb{R}))$ for all $\xi \in \mathbb{R}^n$, where $C^{0,\alpha}(\partial \Omega^h,M_{n}(\mathbb{R}))$ denotes the space of functions of class $C^{0,\alpha}$ from $\partial \Omega^h$ to $M_n(\mathbb{R})$ (cf.~Lanza de Cristoforis \cite[Prop. 6.3]{La07}). We now transform problem \eqref{bvp:nltraceleps} into an integral equation by means of the following (cf.~\cite[Prop.~5.2]{DaMu14}). We find convenient to set $C^{0,\alpha}(\partial {\Omega^h},\mathbb{R}^n)_0\equiv \{f \in C^{0,\alpha}(\partial {\Omega^h},\mathbb{R}^n) \colon \int_{\partial {\Omega^h}}f \, d\sigma=0\}$. \begin{proposition}\label{prop:biju} Let $\omega \in ]1-(2/n),+\infty[$. Let $B\in M_n(\mathbb{R})$. Let $G$ be as in assumption \eqref{assG}. Let $\Lambda$ be the map from $]-\epsilon_0,\epsilon_0[\times C^{0,\alpha}(\partial \Omega^h, \mathbb{R}^n)_0\times \mathbb{R}^n$ to $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)$, defined by \begin{align*} &\Lambda[\epsilon,\theta,\xi](t)\\ &\equiv \frac{1}{2}\theta(t)+w_{\ast}[\omega,\theta](t)+\epsilon^{n-1} \int_{\partial \Omega^h}\sum_{l=1}^n \theta_{l}(s) T(\omega,D R_{n,\omega}^{q,l}(\epsilon(t-s)))\nu_{\Omega^h}(t)\,d\sigma_s\\ &\quad +T(\omega,{Bq^{-1}})\nu_{\Omega^h}(t) -G\Bigl(t,\epsilon v[\omega,\theta](t)\\ &\quad +\epsilon^{n-1} \int_{\partial \Omega^h}R_{n,\omega}^q(\epsilon(t-s))\theta(s)\,d\sigma_s +\epsilon Bq^{-1}t + \xi\Bigr) \quad \forall t \in \partial \Omega^h\,, \end{align*} for all $(\epsilon,\theta,\xi)\in ]-\epsilon_0,\epsilon_0[ \times C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times \mathbb{R}^n$. If $\epsilon \in ]0,\epsilon_0[$, then the map $u[\epsilon,\cdot,\cdot]$ from the set of pairs $(\theta,\xi)\in C^{0,\alpha} (\partial \Omega^h,\mathbb{R}^n)_0 \times \mathbb{R}^n$ that solve the equation \begin{equation} \begin{split}\label{eq:biju1} \Lambda[\epsilon,\theta,\xi]=0 \end{split} \end{equation} to the set of functions $u \in C^{1,\alpha}_{\mathrm{loc}}(\operatorname{cl} \mathbb{S}[{\Omega^h_{p,\epsilon}}]^-,\mathbb{R}^n)$ which solve problem \eqref{bvp:nltraceleps}, which takes $(\theta,\xi)$ to the function defined by \[ u[\epsilon,\theta,\xi](x)\equiv \epsilon^{n-1} \int_{\partial \Omega^h}\Gamma_{n,\omega}^q(x-p-\epsilon s)\theta(s)\,d\sigma_s -Bq^{-1}p+\xi+Bq^{-1}x \] for all $x\in \operatorname{cl}\mathbb{S}[\Omega^h_ {p,\epsilon}]^-$, is a bijection. \end{proposition} Hence we are reduced to analyze equation \eqref{eq:biju1}. To study \eqref{bvp:nltraceleps} for $\epsilon$ small, we first observe that for $\epsilon =0$ we obtain an equation which we address to as the \textit{limiting equation} and which has the form \begin{equation}\label{eq:lim1} \frac{1}{2}\theta(t)+w_{\ast}[\omega,\theta](t)+T(\omega,{Bq^{-1}}) \nu_{\Omega^h}(t)-G(t,\xi)=0 \quad \forall t \in \partial \Omega^h\,. \end{equation} Then we have the following Proposition, which shows, under suitable assumptions, the solvability of the limiting equation (cf.~\cite[Prop.~5.3]{DaMu14}). \begin{proposition}\label{prop:limsys} Let $\omega \in ]1-(2/n),+\infty[$. Let $B\in M_n(\mathbb{R})$. Let $G$ be as in assumption \eqref{assG}. Assume that there exists $\tilde{\xi} \in \mathbb{R}^n$ such that \[ \int_{\partial \Omega^h}G(t,\tilde{\xi})\, d\sigma_t=0\,. \] Then the integral equation \[ \frac{1}{2}\theta(t)+w_{\ast}[\omega,\theta](t)+T(\omega,{Bq^{-1}}) \nu_{\Omega^h}(t)-G(t,\tilde{\xi})=0 \quad \forall t \in \partial \Omega^h \] has a unique solution in $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0$, which we denote by $\tilde{\theta}$. As a consequence, the pair $(\tilde{\theta},\tilde{\xi})$ is a solution in $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times \mathbb{R}^n$ of the limiting equation \eqref{eq:lim1}. \end{proposition} Finally, by a straightforward modification of the proof of \cite[Thm.~5.5]{DaMu14}, we deduce the validity of the following theorem, where we analyze equation \eqref{eq:biju1} around the degenerate value $\epsilon=0$. \begin{theorem}\label{thm:Lmbd} Let $\omega \in ]1-(2/n),+\infty[$. Let $B\in M_n(\mathbb{R})$. Let $G$ be as in assumption \eqref{assG}. Assume that \begin{equation}\label{assFGC1} \text{$F_G $ is a continuosly Fr\'echet differentiable operator from $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)$ to itself.} \end{equation} Assume that there exists $\tilde{\xi} \in \mathbb{R}^n$ such that \begin{equation}\label{assxi} \int_{\partial \Omega^h}G(t,\tilde{\xi})\, d\sigma_t=0\quad\text{and}\quad \det\Big(\int_{\partial \Omega^h}D_uG(t,\tilde{\xi})\, d\sigma_t\Big)\neq0. \end{equation} Let $\Lambda$ be as in Proposition \ref{prop:biju}. Let $\tilde{\theta}$ be the unique function in $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0$ such that $\Lambda[0,\tilde\theta,\tilde{\xi}]=0$ (cf.~Proposition \ref{prop:limsys}). Then there exist $\epsilon_1 \in ]0,\epsilon_0]$, an open neighborhood $\mathcal{U}$ of $(\tilde{\theta},\tilde{\xi})$ in $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n$, and a continuously differentiable map $(\Theta,\Xi)$ from $]-\epsilon_1,\epsilon_1[$ to $\mathcal{U}$, such that the set of zeros of the map $\Lambda$ in $]-\epsilon_1,\epsilon_1[\times\mathcal{U}$ coincides with the graph of $(\Theta,\Xi)$. In particular, $(\Theta[0],\Xi[0])=(\tilde{\theta},\tilde{\xi})$. \end{theorem} \begin{remark}\label{rem:ueps} \rm Let the notation and assumptions of Theorem \ref{thm:Lmbd} hold. Let $u[\cdot,\cdot,\cdot]$ be as in Proposition \ref{prop:biju}. Let $u(\epsilon,x)\equiv u[\epsilon,\Theta[\epsilon],\Xi[\epsilon]](x)$ for all $x \in \operatorname{cl} \mathbb{S}[{\Omega^h_{p,\epsilon}}]^-$ and for all $\epsilon \in ]0,\epsilon_1[$. Then for each $\epsilon \in ]0,\epsilon_1[$ the function $u(\epsilon,\cdot)$ is a solution of problem \eqref{bvp:nltraceleps}. \end{remark} \section{Converging families of solutions}\label{conv} In this section we investigate some limiting and uniqueness properties of converging families of solutions of problem \eqref{bvp:nltraceleps}. \subsection{Preliminary results}\label{potsec} We first need to study some auxiliary integral operators. In the following lemma, we introduce an operator which we denote by $M_\#$. The proof of the lemma can be done by using classical properties of the elastic layer potentials (see, \emph{e.g.}, \cite[Appendix A]{DaLa10a} and Maz'ya \cite[p.~202]{Ma91}). \begin{lemma}\label{Mcanc} Let $\omega \in ]1-(2/n),+\infty[$. Also let $M_\#$ denote the operator from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n$ to $C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$, which takes a pair $(\theta, \xi)$ to the function $M_\#[\theta,\xi]$ defined by \[ M_\#[\theta,\xi](t)\equiv v[\omega,\theta](t)+ \xi \quad \forall t\in\partial\Omega^h\,. \] Then $M_\#$ is a linear homeomorphism. \end{lemma} Then, if $\epsilon \in ]0,\epsilon_0[$, we define the auxiliary integral operator $M_{\epsilon}$ and we prove its invertibility. \begin{lemma}\label{Minv} Let $\omega \in ]1-(2/n),+\infty[$. Let $\epsilon \in ]0,\epsilon_0[$. Let $M_\epsilon$ denote the operator from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n$ to $C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ which takes a pair $(\theta,\xi)$ to the function $M_\epsilon[\theta,\xi]$ defined by \[ M_\epsilon[\theta,\xi](t)\equiv v[\omega,\theta](t)+\epsilon^{n-2}\int_{\partial \Omega^h}R_{n,\omega}^q(\epsilon(t-s))\theta(s)\,d\sigma_s+ \xi\quad \forall t\in\partial\Omega^h\,. \] Then $M_\epsilon$ is a linear homeomorphism. \end{lemma} \begin{proof} We start by proving that $M_\epsilon$ is a Fredholm operator of index $0$. We first note that \[ M_\epsilon [\theta,\xi](t)=M_\#[\theta,\xi](t) +\epsilon^{n-2}\int_{\partial \Omega^h}R_{n,\omega}^q(\epsilon(t-s))\theta(s) \,d\sigma_s\quad \forall t\in\partial\Omega^h\,, \] for all $(\theta,\xi)\in C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0 \times\mathbb{R}^n$. By standard properties of integral operators with real analytic kernels and with no singularity (cf.~\cite[\S 4]{LaMu13a}), we deduce that the linear operator from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0$ to $C^{2,\alpha}(\operatorname{cl}\Omega^h,\mathbb{R}^n)$ which takes $\theta$ to the function $\epsilon^{n-2}\int_{\partial \Omega^h}R_{n,\omega}^q(\epsilon(t-s)) \theta(s)\,d\sigma_s$ of the variable $t\in\operatorname{cl}\Omega^h$ is continuous. Then by the compactness of the imbedding of $C^{2,\alpha}(\operatorname{cl}\Omega^h,\mathbb{R}^n)$ into $C^{1,\alpha}(\operatorname{cl}\Omega^h,\mathbb{R}^n)$, and by the continuity of the trace operator from $C^{1,\alpha}(\operatorname{cl}\Omega^h,\mathbb{R}^n)$ to $C^{1,\alpha}(\partial \Omega^h,\mathbb{R}^n)$, and by Lemma \ref{Mcanc}, we deduce that $M_\epsilon$ is a compact perturbation of the linear homeomorphism $M_\#$, and thus a Fredholm operator of index $0$. Then, by the Fredholm theory, in order to prove that $M_\epsilon$ is a linear homeomorphism, it suffices to show that $M_\epsilon$ is injective. So let $(\theta,\xi) \in C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0 \times\mathbb{R}^n$ be such that $M_\epsilon [\theta,\xi]=0$. Then by the rule of change of variables in integrals, we have \[ M_\epsilon [\theta,\xi](\frac{x-p}{\epsilon})=v_q[\omega, \frac{1}{\epsilon}\theta\bigl(\frac{\cdot - p}{\epsilon}\bigr)](x) +\xi=0 \quad \forall x \in \partial \Omega_{p,\epsilon}^h\, . \] Then by the periodicity of $v_q[\omega,\frac{1}{\epsilon}\theta\bigl(\frac{\cdot - p}{\epsilon}\bigr)]$ and by a straightforward modification of the argument of \cite[Proof of Prop.~4.1]{DaMu14}, we deduce that \[ v_q[\omega,\frac{1}{\epsilon}\theta\bigl(\frac{\cdot - p}{\epsilon}\bigr)](x) +\xi=0 \quad \forall x \in \operatorname{cl}\mathbb{S}[ \Omega_{p,\epsilon}^h]^-\, . \] As a consequence, \[ 0=T(\omega, Dv^-_q[\omega,\frac{1}{\epsilon} \theta\bigl(\frac{\cdot - p}{\epsilon}\bigr)](x))\nu_{\Omega_{p,\epsilon}^h}(x) =\frac{1}{2}\big(\frac{1}{\epsilon}\theta\bigl(\frac{x - p}{\epsilon}\bigr)\big) +w_{q,\ast}[\omega,\frac{1}{\epsilon}\theta\bigl(\frac{\cdot - p}{\epsilon}\bigr)](x) \] for all $x \in \partial \Omega_{p,\epsilon}^h$. Then by \cite[Prop.~4.4]{DaMu14}, we deduce that $\theta=0$ and accordingly $\xi=0$. \end{proof} We can now show that if $\{\varepsilon_{j}\}_{ j\in {\mathbb{N}} }$ is a sequence in $]0,\epsilon_{0}[$ converging to $0$, then $M_{\varepsilon_j}^{(-1)}$ converges to $M_{\#}^{(-1)}$ as $j\to +\infty$. \begin{lemma}\label{MepsMcanc} Let $\omega \in ]1-(2/n),+\infty[$. Let $\{\varepsilon_{j}\}_{ j\in {\mathbb{N}} }$ be a sequence in $]0,\epsilon_{0}[$ converging to $0$. Then $\lim_{j\to+\infty }M_{\varepsilon_j}^{(-1)}=M_{\#}^{(-1)}$ in $\mathcal{L}(C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n), C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n)$. \end{lemma} \begin{proof} Let $N_j$ be the operator from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n$ to $C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ which takes $(\theta,\xi)$ to \[ N_j[\theta,\xi](t)\equiv \varepsilon_j^{n-2} \int_{\partial\Omega^h}R_{n,\omega}^q(\varepsilon_j(t-s))\theta(s)\,d\sigma_s \quad \forall t\in\partial\Omega^h\,,\;\forall j\in\mathbb{N}. \] Let $\mathcal{U}_{\Omega^h}$ be an open bounded neighborhood of $\operatorname{cl}\Omega^h$. Let $\epsilon_\#$ be such that $\epsilon(t-s)\in (\mathbb{R}^n\setminus q\mathbb{Z}^n)\cup\{0\}$ for all $t,s\in\mathcal{U}_{\Omega^h}$ and all $\epsilon\in]-\epsilon_\#,\epsilon_\#[$. By the real analyticity of $R_{n,\omega}^q$ in $(\mathbb{R}^n\setminus q\mathbb{Z}^n)\cup\{0\}$ it follows that the map which takes $(\epsilon,t,s)$ to $R_{n,\omega}^q(\epsilon(t-s))$ is real analytic from $]-\epsilon_\#,\epsilon_\#[\times\mathcal{U}_{\Omega^h}\times\mathcal{U}_{\Omega^h}$ to $M_n(\mathbb{R})$. Hence, there exists a real analytic map $\tilde{R}_{n,\omega}^q$ from $]-\epsilon_\#,\epsilon_\#[\times\mathcal{U}_{\Omega^h}\times\mathcal{U}_{\Omega^h}$ to $M_n(\mathbb{R})$ such that $R_{n,\omega}^q(\epsilon (t-s))-R_{n,\omega}^q(0) =\epsilon \tilde{R}_{n,\omega}^q(\epsilon,t,s)$ for all $t,s\in\mathcal{U}_{\Omega^h}$ and all $\epsilon\in]-\epsilon_\#,\epsilon_\#[$. Then, by the membership of $\theta$ in $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0$, one has \[ N_j[\theta,\xi](t)= \varepsilon_j^{n-1}\int_{\partial\Omega^h} \tilde{R}_{n,\omega}^q(\varepsilon_j,t,s)\theta(s)\,d\sigma_s\quad \forall t\in\partial\Omega^h \] for all $j$ such that $\varepsilon_j\in]0,\epsilon_\#[$ and for all $(\theta,\xi)\in C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n$. Then, by standard properties of integral operators with real analytic kernels and with no singularities (cf. \cite[\S 4]{LaMu13a}), we deduce that $\lim_{j\to+\infty}N_j=0$ in $\mathcal{L}(C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0 \times\mathbb{R}^n, C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n))$. Since $M_{\varepsilon_j}=M_\#+N_j$, it follows that $\lim_{j\to+\infty}M_{\varepsilon_j}=M_\sharp$ in $\mathcal{L}(C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n, C^{1,\alpha}(\partial\Omega^h, \mathbb{R}^n))$. Then by the continuity of the mapping from the open subset of the invertible operators of $\mathcal{L}(C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n, C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n))$ to $\mathcal{L}(C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n), C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n)$ which takes an operator to its inverse, one deduces that $\lim_{j\to+\infty }M_{\varepsilon_j}^{(-1)}=M_{\#}^{(-1)}$ (cf. e. g., Hille and Phillips \cite[Thms. 4.3.2 and 4.3.3]{HiPh57}). \end{proof} \subsection{Limiting behavior of a converging family of solutions} We are now ready to investigate in this subsection the limiting behavior of a converging family of solutions of problem \eqref{bvp:nltraceleps}. To begin with, in the following proposition we consider the limiting behavior of converging families of $q$-periodic displacement functions. \begin{proposition}\label{periodic} Let $\omega \in ]1-(2/n),+\infty[$. Let $\{\varepsilon_{j}\}_{ j\in {\mathbb{N}} }$ be a sequence in $]0,\epsilon_{0}[$ converging to $0$ and let $\{u_{\#,j}\}_{j\in{\mathbb{N}} }$ be a sequence of functions such that for each $j \in \mathbb{N}$ \begin{gather*} u_{\#,j}\in C^{1,\alpha}_{\mathrm{loc}}( \operatorname{cl}{\mathbb{S}}[\Omega_{p,\varepsilon_{j}}^h]^{-},\mathbb{R}^n)\,,\quad u_{\#,j} \text{ is $q$-periodic},\\ \text{and } \operatorname{div} T(\omega, Du_{\#,j} )=0\text{ in } \mathbb{S}[\Omega_{p,\varepsilon_{j}}^h]^{-}\,. \end{gather*} Assume that there exists a function $v_\#\in C^{1,\alpha}(\partial\Omega^h, \mathbb{R}^n)$ such that \begin{equation}\label{ujvcanc} \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j\cdot)\big|_{\partial\Omega^h} =v_\#\quad\text{in }C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)\,. \end{equation} Then there exists a pair $(u_\#,\xi_\#)\in C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^n \setminus\Omega^h,\mathbb{R}^n)\times\mathbb{R}^n$ such that \[ v_\#=u_{\#}\big|_{\partial\Omega^h}+\xi_\#,\quad \operatorname{div} T(\omega, Du_\# )=0\text{ in } \mathbb{R}^n\setminus\operatorname{cl}\Omega^h\,, \] and such that \[ \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-2+\delta_{2,n}}|u_\#(x)| < \infty\, , \quad \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-1 +\delta_{2,n}}|D u_\#(x)| < \infty\,. \] Moreover, \begin{equation}\label{ujlim} \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j\cdot) \big|_{\operatorname{cl} \mathcal{O}} =u_{\#}\big|_{\operatorname{cl}\mathcal{O}} +\xi_\#\quad\text{in }C^{1,\alpha}(\operatorname{cl} \mathcal{O},\mathbb{R}^n) \end{equation} for all open bounded subsets $\mathcal{O}$ of $\mathbb{R}^n \setminus \operatorname{cl}\Omega^h$, and \begin{equation}\label{ujlimbis} \lim_{j\to+\infty} u_{\#,j}\big|_{\operatorname{cl}\tilde{\mathcal{O}}} =\xi_\#\quad\text{in }C^{k}(\operatorname{cl}\tilde{\mathcal{O}},\mathbb{R}^n) \end{equation} for all $k \in \mathbb{N}$ and for all open bounded subsets $\tilde{\mathcal{O}}$ of $\mathbb{R}^n$ such that $\operatorname{cl}\tilde{\mathcal{O}} \subseteq \mathbb{R}^n \setminus (p+q \mathbb{Z}^n)$. \end{proposition} \begin{proof} Let \[ (\theta_j,\xi_j)\equiv M^{(-1)}_{\varepsilon_j} [u_{\#,j}(p+\varepsilon_j \cdot)\big|_{\partial \Omega^h}] \] for all $j\in\mathbb{N}$ and $(\theta_\#,\xi_\#)\equiv M^{(-1)}_{\#}[v_{\#}]$. Since the evaluation mapping from ${\mathcal{L}}(C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n), C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_{0}\times{\mathbb{R}}^n)\times C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_{0}\times{\mathbb{R}^n}$, which takes a pair $(A,v)$ to $A[v]$ is bilinear and continuous, the limiting relation \eqref{ujvcanc} and Lemma \ref{MepsMcanc} imply that \begin{equation}\label{tcjtccanc} \lim_{j\to+\infty}(\theta_j,\xi_j) =\lim_{j\to+\infty}M^{(-1)}_{\varepsilon_j}[u_{\#,j}(p+\varepsilon_j \cdot)\big|_{\partial \Omega^h}]=M^{(-1)}_{\#}[v_{\#}]=(\theta_\#,\xi_\#) \end{equation} in $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)_{0}\times{\mathbb{R}^n}$. Also, one has \begin{equation}\label{ujrep0} u_{\#,j}(x)={\varepsilon^{n-2}_j}\int_{\partial\Omega^h} \Gamma_{n,\omega}^q(x-p-\varepsilon_j s)\theta_j(s)\,d\sigma_s +\xi_j\quad\forall x\in\operatorname{cl}\mathbb{S}[\Omega_{p,\varepsilon_j}^h]^-\,, \forall j \in \mathbb{N}\,. \end{equation} Then one has \begin{equation}\label{ujrep} u_{\#,j}(p+\varepsilon_j t)=v[\omega,\theta_j](t)+{\varepsilon^{n-2}_j} \int_{\partial\Omega^h} R_{n,\omega}^q(\varepsilon_j(t- s))\theta_j(s)\,d\sigma_s +\xi_j \end{equation} for all $t\in\mathbb{R}^n\setminus\cup_{z\in\mathbb{Z}^n} (\varepsilon_j^{-1}qz+\operatorname{cl}\Omega^h)$. By continuity of the map from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to $C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ which takes $\theta$ to $v[\omega,\theta]\big|_{\partial\Omega^h}$, by standard properties of integral operators with real analytic kernels and with no singularities (cf.~\cite[\S 4]{LaMu13}), by condition $\int_{\partial\Omega^h}\theta_\# d\sigma=0$, and by \eqref{tcjtccanc}, one verifies that \[ \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j \cdot)\big|_{\partial\Omega^h} =v[\omega,\theta_\#]\big|_{\partial\Omega^h}+\xi_\#\quad \text{in }C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)\,. \] Hence, the limiting relation in \eqref{ujvcanc} implies that $v_\#=v[\omega,\theta_\#]\big|_{\partial\Omega^h}+\xi_\#$. Now the validity of the proposition follows by setting $u_\#(t)\equiv v[\omega,\theta_\#](t)$ for all $t\in\mathbb{R}^n\setminus\Omega^h$. Indeed, by classical results for elastic layer potentials and by condition $\int_{\partial\Omega^h}\theta_\# d\sigma=0$, we have $u_\# \in C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^n\setminus\Omega^h,\mathbb{R}^n)$, $\operatorname{div} T(\omega, Du_\# )=0$ in $\mathbb{R}^n\setminus\operatorname{cl}\Omega^h$, and \[ \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-2+\delta_{2,n}}|u_\#(x)| < \infty\, , \quad \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-1+\delta_{2,n}}|D u_\#(x)| < \infty\, . \] Finally, the validity of \eqref{ujlim} for all open bounded subsets $\mathcal{O}$ of $\mathbb{R}^n \setminus \operatorname{cl}\Omega^h$ follows by equality \eqref{ujrep}, by the limiting relation in \eqref{tcjtccanc}, by the continuity of the map from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to $C^{1,\alpha}(\operatorname{cl}\mathcal{O},\mathbb{R}^n)$ which takes $\theta$ to $v[\omega,\theta]\big|_{\operatorname{cl}\mathcal{O}}$, by standard properties of integral operators with real analytic kernels and with no singularities, and by $\int_{\partial\Omega^h}\theta_\# d\sigma=0$. Similarly, the validity of \eqref{ujlimbis} for all $k \in \mathbb{N}$ and for all open bounded subsets $\tilde{\mathcal{O}}$ of $\mathbb{R}^n$ such that $\operatorname{cl}\tilde{\mathcal{O}} \subseteq \mathbb{R}^n \setminus (p+q \mathbb{Z}^n)$ follows by equality \eqref{ujrep0}, by the limiting relation in \eqref{tcjtccanc}, by standard properties of integral operators with real analytic kernels and with no singularities (cf.~\cite[\S 4]{LaMu13}), and by $\int_{\partial\Omega^h}\theta_\# d\sigma=0$. \end{proof} We are now ready to prove the main result of this subsection, where we study the limiting behavior of converging families of solutions of problem \eqref{bvp:nltraceleps}. \begin{theorem} \label{thm:rem} Let $\omega \in ]1-(2/n),+\infty[$. Let $B\in M_n(\mathbb{R})$. Let $G\in C^{0}(\partial\Omega^h\times {\mathbb{R}}^n,\mathbb{R}^n)$ be such that $F_{G }$ is continuous from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to itself. Let $\{\varepsilon_{j}\}_{ j\in {\mathbb{N}} }$ be a sequence in $]0,\epsilon_{0}[$ converging to $0$ and let $\{u_{j}\}_{j\in {\mathbb{N}} }$ be a sequence of functions such that $u_{j}$ belongs to $C^{1,\alpha}_{\mathrm{loc}}( \operatorname{cl}{\mathbb{S}} [\Omega_{p,\varepsilon_{j}}^h]^{-},\mathbb{R}^n)$ and is a solution of \eqref{bvp:nltraceleps} with $\epsilon=\varepsilon_{j}$ for all $j \in \mathbb{N}$. Assume that there exists a function $v_\ast\in C^{1,\alpha}(\partial\Omega^h, \mathbb{R}^n)$ such that \[ \lim_{j\to+\infty} u_{j}(p+\varepsilon_j\cdot)\big|_{\partial\Omega^h}=v_\ast \quad \text{in $C^{1,\alpha}(\partial\Omega^h, \mathbb{R}^n)$}\, . \] Then there exists $\xi_\ast \in \mathbb{R}^n$ such that \[ v_\ast=\xi_\ast\text{ on }\partial\Omega^h , \quad \int_{\partial \Omega^h}G(t,\xi_\ast)\, d\sigma_t=0\,. \] Moreover, \[ \lim_{j\to+\infty} u_{j}(p+\varepsilon_j\cdot)\big|_{\operatorname{cl} \mathcal{O}}=\xi_\ast \quad\text{in } C^{1,\alpha}(\operatorname{cl}\mathcal{O},\mathbb{R}^n) \] for all open bounded subsets $\mathcal{O}$ of $\mathbb{R}^n \setminus \operatorname{cl}\Omega^h$, and \[ \lim_{j\to+\infty} u_{j}\big|_{\operatorname{cl}\tilde{\mathcal{O}}} =\xi_\ast+Bq^{-1}(\cdot-p)\quad\text{in }C^{k}(\operatorname{cl} \tilde{\mathcal{O}},\mathbb{R}^n) \] for all $k \in \mathbb{N}$ and for all open bounded subsets $\tilde{\mathcal{O}}$ of $\mathbb{R}^n$ such that $\operatorname{cl}\tilde{\mathcal{O}} \subseteq \mathbb{R}^n \setminus (p+q \mathbb{Z}^n)$. \end{theorem} \begin{proof} We set \begin{gather*} u_{\#,j}(x) \equiv u_j(x)-B q^{-1}x \quad \forall x \in \operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}^h]^{-}\,,\quad \forall j\in \mathbb{N}\, ,\\ v_{\#}(x) \equiv v_{\ast}(x)- B q^{-1} p \quad \forall x \in \partial\Omega^h\,. \end{gather*} Then for each $j \in \mathbb{N}$, the function $u_{\#,j}\in C^{1,\alpha}_{\mathrm{loc}}( \operatorname{cl}{\mathbb{S}}[\Omega_{p,\varepsilon_{j}}^h]^{-},\mathbb{R}^n)$, $u_{\#,j}$ is $q$-periodic and $\operatorname{div} T(\omega, Du_{\#,j} )=0$ in $\mathbb{S}[\Omega_{p,\varepsilon_{j}}^h]^{-}$. We have \[ v_\#\in C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)\, , \quad \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j\cdot)\big|_{\partial\Omega^h} =v_\#\quad\text{in }C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)\,. \] Hence, by Proposition \ref{periodic}, there exists a pair $(u_\#,\xi_\#)\in C^{1,\alpha}_{\mathrm{loc}} (\mathbb{R}^n\setminus\Omega^h,\mathbb{R}^n)\times\mathbb{R}^n$ such that \begin{gather} v_\#=u_{\#}\big|_{\partial \Omega^h}+\xi_\#\, , \quad \operatorname{div} T(\omega, Du_\# )=0 \quad \text{in $\mathbb{R}^n\setminus\operatorname{cl}\Omega^h$}\, ,\label{eq:remproof0}\\ \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-2+\delta_{2,n}}|u_\#(x)| < \infty\, , \quad \sup _{x \in \mathbb{R}^n \setminus \Omega^h}|x|^{n-1+\delta_{2,n}}|D u_\#(x)| < \infty\, .\label{eq:remproof0a} \end{gather} Moreover, \begin{equation}\label{eq:remproof0b} \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j\cdot)\big|_{\operatorname{cl} \mathcal{O}}=u_{\#}\big|_{\operatorname{cl} \mathcal{O}}+\xi_\#\quad\text{in }C^{1,\alpha} (\operatorname{cl}\mathcal{O},\mathbb{R}^n) \end{equation} for all open bounded subsets $\mathcal{O}$ of $\mathbb{R}^n \setminus \operatorname{cl}\Omega^h$, and \[ \lim_{j\to+\infty} u_{\#,j}\big|_{\operatorname{cl}\tilde{\mathcal{O}}}=\xi_\#\quad \text{in }C^{k}(\operatorname{cl}\tilde{\mathcal{O}},\mathbb{R}^n) \] for all $k \in \mathbb{N}$ and for all open bounded subsets $\tilde{\mathcal{O}}$ of $\mathbb{R}^n$ such that $\operatorname{cl}\tilde{\mathcal{O}} \subseteq \mathbb{R}^n \setminus (p+q \mathbb{Z}^n)$. Then we observe that \begin{equation}\label{eq:remproof1} T (\omega, D u_{\#,j} (p+\varepsilon_j t) +B q^{-1})\nu_{\Omega_{p,\varepsilon_j}^h}(p+\varepsilon_j t) =G(t,u_{\#,j}(p+\varepsilon_j t)+Bq^{-1}(p+\varepsilon_j t)) \end{equation} for all $t\in\partial\Omega^h$ and all $j\in\mathbb{N}$, which implies \begin{equation}\label{eq:remproof1a} \begin{aligned} &T \Bigl (\omega, D_t \bigl (u_{\#,j} (p+\varepsilon_j t)\bigr) \Bigr) \nu_{\Omega^h}(t)\\ &=-\varepsilon_j T(\omega,Bq^{-1})\nu_{\Omega^h}(t) +\varepsilon_j G(t,u_{\#,j}(p+\varepsilon_j t)+Bq^{-1}p +\varepsilon_j B q^{-1}t) \end{aligned} \end{equation} for all $t\in\partial\Omega^h$, and all $j\in\mathbb{N}$. Then, by \eqref{eq:remproof0b}, by the continuity of $F_G$ from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to itself, and by taking the limit as $j\to+\infty$ in \eqref{eq:remproof1a}, one obtains \[ T (\omega, D u_{\#} (t))\nu_{\Omega^h}(t)=0 \quad \forall t\in\partial\Omega^h\, , \] which, together with \eqref{eq:remproof0} and \eqref{eq:remproof0a}, implies $u_\#=0$. In particular, \begin{equation}\label{eq:remproof1b} \lim_{j\to+\infty} u_{\#,j}(p+\varepsilon_j\cdot)\big|_{\operatorname{cl} \mathcal{O}}=\xi_\#\quad\text{in } C^{1,\alpha}(\operatorname{cl}\mathcal{O},\mathbb{R}^n) \end{equation} for all open bounded subsets $\mathcal{O}$ of $\mathbb{R}^n \setminus \operatorname{cl}\Omega^h$. Furthermore, by \eqref{eq:remproof1}, by \cite[Prop.~4.2]{DaMu14}, and by the equality $\int_{\partial \Omega_{p,\varepsilon_j}^h}T (\omega, Bq^{-1})\nu_{\Omega_{p,\varepsilon_j}^h}(x)\, d\sigma_x=0$, one has \begin{equation}\label{eq:remproof2} \begin{aligned} 0&=\frac{1}{\varepsilon_j^{n-1}}\int_{\partial \Omega_{p,\varepsilon_j}^h} T (\omega, D u_{\#,j} (x)+Bq^{-1})\nu_{\Omega_{p,\varepsilon_j}^h}(x)\, d\sigma_x\\ &= \int_{\partial \Omega^h}G(t,u_{\#,j}(p+\varepsilon_j t)+Bq^{-1}p +\varepsilon_j B q^{-1}t)\, d\sigma_t \end{aligned} \end{equation} for all $j \in \mathbb{N}$. Then, by the continuity of $F_G$ from $C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ to itself, by the limiting relation in \eqref{eq:remproof1b}, and by letting $j\to+\infty$ in \eqref{eq:remproof2}, one deduces \[ \int_{\partial \Omega^h}G(t,\xi_\#+Bq^{-1}p)\, d\sigma_t=0\,. \] Finally, by setting $\xi_\ast \equiv \xi_\#+Bq^{-1}p$, the validity of the theorem follows. \end{proof} \subsection{A local uniqueness result for converging families of solutions} In this subsection we prove that a converging family of solutions of \eqref{bvp:nltraceleps} is essentially unique in a local sense which we clarify in the following theorem. \begin{theorem}\label{localuniq} Let $\omega \in ]1-(2/n),+\infty[$. Let $B\in M_n(\mathbb{R})$. Let $G$ be as in assumptions \eqref{assG}, \eqref{assFGC1}. Let $\{\varepsilon_{j}\}_{ j\in {\mathbb{N}} }$ be a sequence in $]0,\epsilon_{0}[$ converging to $0$. Let $\{u_{j}\}_{j\in {\mathbb{N}} }$ and $\{v_{j}\}_{j\in {\mathbb{N}} }$ be sequences such that $u_{j}$ and $v_j$ belong to $C^{1,\alpha}_{\mathrm{loc}}( \operatorname{cl}{\mathbb{S}} [\Omega_{p,\varepsilon_{j}}^h]^{-},\mathbb{R}^n)$ and both $u_{j}$ and $v_j$ are solutions of \eqref{bvp:nltraceleps} with $\epsilon=\varepsilon_{j}$ for all $j\in\mathbb{N}$. Assume that there exists a function $v_\ast\in C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n)$ such that \begin{equation}\label{localuniq.eq0} \lim_{j\to+\infty} u_{j}(p+\varepsilon_j\cdot)\big|_{\partial\Omega^h} =\lim_{j\to+\infty} v_{j}(p+\varepsilon_j\cdot)\big|_{\partial\Omega^h} =v_\ast\quad\text{in }C^{1,\alpha}(\partial\Omega^h,\mathbb{R}^n) \end{equation} and that \[ \det\Big(\int_{\partial\Omega^h} D_{u}G(t,v_\ast(t))\,d\sigma_t\Big)\neq 0\,. \] Then there exists a natural number $j_0\in\mathbb{N}$ such that $u_j=v_j$ for all $j\ge j_0$. \end{theorem} \begin{proof} We first observe that the family $\{u_j\}_{j\in\mathbb{N}}$ and the function $v_\ast$ satisfy the conditions in Theorem \ref{thm:rem}. As a consequence, there exists $\tilde{\xi} \in\mathbb{R}^n$ such that \[ \lim_{j\to+\infty}u_{j}(p+\varepsilon_{j}\cdot)\big|_{\partial\Omega^h} = \tilde{\xi}\quad{\text{in}}\ C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)\, . \] Then by \eqref{localuniq.eq0} one also has \[ \lim_{j\to+\infty}v_{j}(p+\varepsilon_{j}\cdot)\big|_{\partial\Omega^h} = \tilde{\xi}\quad{\text{in}}\ C^{0,\alpha}(\partial\Omega^h,\mathbb{R}^n)\, . \] Moreover, we deduce that $\tilde{\xi}$ satisfies assumption \eqref{assxi}. By Proposition \ref{prop:biju}, for each $j \in \mathbb{N}$ there exist and are unique two pairs $(\theta_{1,j},\xi_{1,j})$, $(\theta_{2,j},\xi_{2,j})$ in $C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times \mathbb{R}^n$ such that \begin{equation}\label{eq:uniqueps9a} u_j(x)=u[\varepsilon_j,\theta_{1,j},\xi_{1,j}](x)\,, \quad v_j(x)=u[\varepsilon_j,\theta_{2,j},\xi_{2,j}](x)\, , \quad \forall x \in \operatorname{cl}\mathbb{S}[\Omega^h_{p,\varepsilon_j}]^{-}\,. \end{equation} Let $\tilde{\theta}$, $\epsilon_1$ be as in Theorem \ref{thm:Lmbd}. Then to show the validity of the theorem, it will be enough to prove that \begin{gather} \lim_{j \to +\infty}(\theta_{1,j},\xi_{1,j}) =(\tilde{\theta},\tilde{\xi}) \quad \text{in } C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n\,, \label{eq:uniqueps4a}\\ \lim_{j \to +\infty}(\theta_{2,j},\xi_{2,j}) =(\tilde{\theta},\tilde{\xi}) \quad \text{in } C^{0,\alpha}(\partial \Omega^h,\mathbb{R}^n)_0\times\mathbb{R}^n\,. \label{eq:uniqueps4b} \end{gather} Indeed, if we denote by $\mathcal{U}$ the neighborhood of Theorem \ref{thm:Lmbd}, the limiting relations in \eqref{eq:uniqueps4a}, \eqref{eq:uniqueps4b} imply that there exists $j_0 \in \mathbb{N}$ such that $(\varepsilon_j,\theta_{1,j},\xi_{1,j}), (\varepsilon_j,\theta_{2,j},\xi_{2,j}) \in ]0,\epsilon_1[\times \mathcal{U}$ for all $j \geq j_0$ and thus Theorem \ref{thm:Lmbd} implies that $(\theta_{1,j},\xi_{1,j})=(\theta_{2,j},\xi_{2,j}) =(\Theta[\varepsilon_j],\Xi[\varepsilon_j])$ for all $j \geq j_0$, and accordingly the theorem follows by \eqref{eq:uniqueps9a}. The proof of the limits in \eqref{eq:uniqueps4a}, \eqref{eq:uniqueps4b} follows the lines of \cite[Proof of Thm.~7.1]{DaMu14} and is accordingly omitted. \end{proof} \subsection*{Acknowledgments} M.~Dalla Riva was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (``FCT--Funda{\c c}{\~a}o para a Ci\^encia e a Tecnologia''), within project PEst-OE/MAT/UI4106/2014. M.~Dalla Riva was also supported by the Portuguese Foundation for Science and Technology FCT with the research grant SFRH/BPD/64437/2009. P.~Musolino is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). M.~Dalla Riva and P.~Musolino were also supported by ``Progetto di Ateneo: Singular perturbation problems for differential operators -- CPDA120171/12" - University of Padova. 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