$(20$ points$)$ Let $$ be a region in ${}^2$ and let its boundary $$ be decomposed into $4$ non$-$overlapping boundary segments such that $\frac{?}{b}=ar({}_1{}_2{}_3{}_4).$ Let $n$ be an outward unit normal to $$ and define the axis $s$ such that $s$ and $n$ form a right hand basis $($see figure$).$ Consider the following strong form statement of the linear elastostatics for this problem.

Given $f_i$:$$;$g_i$:${}_1$;$h_i$:${}_2$;$g_n$ and $h_s$:${}_3$; and $g_s$ and $h_n$:${}_4$; find $u_i\frac{\text{:}}{b}ar()$ such that

${}_{ij,j}+f_i=0,in$

subject to

$u_i=g_{i}on{}_1$

${}_{ij}{n}_j=h_{i}on{}_2$

$u_i{n}_i=g_n$ and ${}_{ij}{n}_j{s}_i=h_{s}on{}_3$

$u_i{s}_i=g_s$ and ${}_{ij}{n}_j{n}_i=h_{n}on{}_4$

where ${}_{ij}=C_{\text{i}\text{j}\text{k}\text{l}}{u}_{(k,l)}.$

Establish the Galerkin weak form for this problem in which the $g-$type boundary conditions are treated as essential boundary conditions and all the $h-$type boundary conditions are treated as natural boundary conditions. State all the requirements on the spaces $$ and $V.$ Hint: $w=w_{n}n+w_{s}s,$ and $w_i=w_n{n}_i+w_s{s}_i$