Mixed boundary value transmission problems for the linear theory of elasticity

In previous works, the author studied the general approach to solving mixed boundary,spectral, and initial-boundary transmission problems. In this paper, this approach is applied to the mixed boundary transmission problem of the linear theory of elasticity. On the basis of the corresponding Green’s formulas, the solution of the problem can be represented as the sum of the solutions of auxiliary problems.

Let us consider an elastic body consisting of two joined regions $\Omega_1$ and $\Omega_2$ from $\mathbb{R}^{m}$. Let its outer boundaries be Lipschitz $(\Gamma_{11}\cup\Gamma_{22})\cup\partial\Gamma_{11}=\Gamma=\partial\Omega$, moreover, the contour $\partial\Gamma_{11}$ is also Lipschitz. We suppose that the interface $\Gamma_{12}=\Gamma_{21}$ is Lipschitz.

Let us consider an elastic body consisting of two joined regions $\Omega_1$ and $\Omega_2$ from $\mathbb{R}^{m}$. Let its outer boundaries be Lipschitz $(\Gamma_{11}\cup\Gamma_{22})\cup\partial\Gamma_{11}=\Gamma=\partial\Omega$, moreover, the contour $\partial\Gamma_{11}$ is also Lipschitz. We suppose that the interface $\Gamma_{12}=\Gamma_{21}$ is Lipschitz.

\begin{equation}
\begin{split}
L_1\overrightarrow{v_1} &= -[\mu_1\vartriangle{v}+(\lambda_1+\mu_1)\triangledown\,{div}\,\overrightarrow{v_1}]=\overrightarrow{f_1}(x)(в\;\Omega_1),\\
L_2\overrightarrow{v_2} &= -[\mu_2\vartriangle{v}+(\lambda_2+\mu_2)\triangledown\,{div}\,{\overrightarrow{v_2}}]=\overrightarrow{f_2}(x)(в\;\Omega_2);
\end{split}\tag{1}\label{eq:1}
\end{equation}
\begin{equation}
\gamma_{11}\overrightarrow{v_1}=\overrightarrow{\varphi}_1 (на\;\Gamma_{11}),\;\;\gamma_{22}\overrightarrow{v_2}=\overrightarrow{\varphi}_2 (на\;\Gamma_{22});
\tag{2}\label{eq:2}
\end{equation}
\begin{equation}
\gamma_{21}\overrightarrow{v_1}-\gamma_{12}\overrightarrow{v_2}=\overrightarrow{\varphi}_{21},\;\; P_{21}\overrightarrow{v_1}+P_{12}\overrightarrow{v_2}=\overrightarrow{\psi}_{21} (на\;\Gamma_{12}=\Gamma_{21}),
\tag{3}\label{eq:3}
\end{equation}
\begin{equation*}
P_{21}\overrightarrow{v_1}:=\sum_{j,k=1}^3(\mu_1\tau_{jk}(\overrightarrow{v_1})+\lambda_1\delta_{jk}\,{div}\,{\overrightarrow{v_1}})\cos(\overrightarrow{v_1},\overrightarrow{e_j})\overrightarrow{e_j},
\end{equation*}
\begin{equation*}
P_{12}\overrightarrow{v_2}:=\sum_{j,k=1}^3(\mu_2\tau_{jk}(\overrightarrow{v_2})+\lambda_2\delta_{jk}\,{div}\,{\overrightarrow{v_2}})\cos(\overrightarrow{v_2},\overrightarrow{e_j})\overrightarrow{e_j}.
\end{equation*}

On the basis of the corresponding Green’s formulas
\begin{equation}
(\overrightarrow{\eta}, \overrightarrow{v})_{\overrightarrow{H}^1(\Omega)}=\langle\overrightarrow{\eta},L\overrightarrow{v}\rangle_{\overrightarrow{L}_2(\Omega)}+\sum_{k=1}^l\langle\gamma_k\overrightarrow{\eta},P_k\overrightarrow{v}\rangle_{\overrightarrow{L}_2(\Gamma_k)}, \;\forall\overrightarrow{\eta}, \overrightarrow{v}\in \widehat{\overrightarrow{H}}^1(\Omega),
\tag{4}\label{eq:4}
\end{equation}
\begin{equation*}
\gamma_k\overrightarrow{\eta}=\overrightarrow{\eta}\arrowvert_{\Gamma_k} \in \widetilde{\overrightarrow{H}}^{1/2}(\Gamma), P_k\overrightarrow{v}:=(P\overrightarrow{v})\arrowvert_{\Gamma_k} \in \overrightarrow{H}^{-1/2}(\Gamma), k=1,2;
\end{equation*}

\begin{equation}
(\overrightarrow{\eta}, \overrightarrow{v})_{\overrightarrow{H}^1(\Omega)}=\langle\overrightarrow{\eta},L\overrightarrow{v}\rangle_{\overrightarrow{L}_2(\Omega)}+\sum_{k=1}^l\langle\gamma_k\overrightarrow{\eta},P_k\overrightarrow{v}\rangle_{\overrightarrow{L}_2(\Gamma_k)}, \;\forall\overrightarrow{\eta}, \overrightarrow{v}\in \check{\overrightarrow{H}}^1_{\Gamma_1}(\Omega),
\tag{5}\label{eq:5}
\end{equation}
\begin{equation*}
\gamma_k\overrightarrow{\eta}=\overrightarrow{\eta}\arrowvert_{\Gamma_k} \in \overrightarrow{H}^{1/2}(\Gamma), P_k\overrightarrow{v}:=(P\overrightarrow{v})\arrowvert_{\Gamma_k} \in \widetilde{\overrightarrow{H}}^{-1/2}(\Gamma), k=1,2,
\end{equation*}
the solution of the problem can be represented as the sum of the solutions of auxiliary problems.
The traces $\gamma_k\overrightarrow{v} \in \overrightarrow{H}^{1/2}(\Gamma)$ have restrictions on $\Gamma_k$, that can be continued by zero in the class of $\overrightarrow{H}^{1/2}(\Gamma)$. Wherein $\overrightarrow{v}$ belong to the space $\widehat{\overrightarrow{H}}^1_{\Gamma_1}(\Omega)$.
Functions $P_k\overrightarrow{v}$ in the second formula can be continued by zero on $\Gamma$ in the class of $\overrightarrow{H}^{-1/2}(\Gamma)$. Wherein $\overrightarrow{v}$ belong to the space $\check{\overrightarrow{H}}^1_{\Gamma_1}(\Omega)$, that is, the space of functions whose $P_k\overrightarrow{v}$ extendible by zero.
For the four auxiliary problems, we find weak solutions using Green’s formulas. Original problem (\ref{eq:1})-(\ref{eq:3}) is the sum of the auxiliary solutions.

Keywords: Green’s formula, transmission problem, weak solution, Lipschitz boundaries, theory of elasticity.