I want to prove the existence of the solution of this system by using the FaedoGalerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I suggest the basis of the following spaces : \begin{equation} H_{\Gamma _{0}}^{1}(\Omega )=\left\{ u\in H^{1}(\Omega );\text{ }u=0\right\} , \end{equation} and $$H_0(\Gamma_1)$$ to deal with the internal and boudary termes. Is this write? thanx. \begin{equation} \left\{ \begin{array}{rrrr} u_{tt}\Delta u=0,&\text{in} & \Omega \times ]0,T[ & \left( 1.1\right) \\ u=0, & \text{on } & \Gamma _{0}\times ]0,T[ & \left( 1.2\right) \\ uw=0, & \text{on} & \Gamma _{1}\times ]0,T[ & \left( 1.3\right) \\ w_{tt}\Delta _{T}w+\partial _{\nu }u+w_{t}=0, & \text{on} & \Gamma _{1}\times ]0,T[ & \left( 1.4\right) \\ w=0,& \text{on } & \partial \Gamma _{1}\times ]0,T[ & \left( 1.5\right) \\ u(.,0)=u_{0},\text{ \ }u_{t}(.,0)=u_{1} & \text{in} & \Omega & \left( 1.6\right) \\ w(.,0)=w_{0},\text{ \ }w_{t}(.,0)=w_{1}\text{\ } & \text{on} & \Gamma _{1} & \left( 1.7\right) \end{array}% \right. \label{E1} \end{equation}
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1$\begingroup$ Additionally, you will have to explain the notation $\Delta_T$ and also the slashes that appears in equations (1.6) and (1.7) (are they just LaTeX errors?). Are $\Gamma_0$ and $\Gamma_1$ parts of the boundaries? (Is $\Delta_T$ the Laplacian restricted to the boundary surface?) What is meant by $\partial\Gamma_1$ in equation (1.5)? $\endgroup$ – Willie Wong Feb 14 '18 at 16:43

$\begingroup$ Yes sir. ${\Delta _T}$ is the LaplaceBeltrami operetor on ${\Gamma _1}$, ${\Gamma _0}$ and ${\Gamma _1}$ are non empty parts of $\Omega $. I think that I should define another space ${H^1}({\Gamma _1})$. $\endgroup$ – Gustave Feb 15 '18 at 9:34