= Solution
Let the <boundary> split into a displacement part $\Gamma_D$ and a <traction> part $\Gamma_N$. The linearized stored <elastic energy> functional is
$$
\boxed{I^{\mathrm{lin}}[u]=\frac12\int_\Omega\mathbb A_{iA jB}u_{i,A}u_{j,B}\,dx.}
$$
Its corresponding total potential subtracts the work of fixed body and surface loads:
$$
\Pi[u]=I^{\mathrm{lin}}[u]-\int_\Omega f\cdot u\,dx-\int_{\Gamma_N}g\cdot u\,dA.
$$
The problem in the <calculus of variations> is to minimize $\Pi$ among admissible displacements with $u=u_D$ on $\Gamma_D$. For a stable quadratic <elastic energy> this is the usual energy minimization formulation. Its stationarity condition suffices to derive equilibrium.
Vary by $u+sv$, with $v=0$ on $\Gamma_D$. Major <symmetry> combines the two differentiated quadratic terms, giving
$$
\int_\Omega S^{\mathrm{lin}}:Dv\,dx=\int_\Omega f\cdot v\,dx+\int_{\Gamma_N}g\cdot v\,dA,
\qquad S^{\mathrm{lin}}_{iA}=\mathbb A_{iA jB}u_{j,B}.
$$
<Integration by parts>, first with test functions compactly supported in the interior, gives $-\operatorname{Div}S^{\mathrm{lin}}=f$. Then arbitrary traces on $\Gamma_N$ give the natural <traction> condition. The complete <boundary value problem> is
$$
\boxed{-\operatorname{Div}S^{\mathrm{lin}}=f\text{ in }\Omega,\qquad
S^{\mathrm{lin}}N=g\text{ on }\Gamma_N,\qquad u=u_D\text{ on }\Gamma_D.}
$$
For a <homogeneous hyperelastic material> the <elasticity tensor> is constant. In <isotropic> <linear elasticity> the <constitutive equation> is $S^{\mathrm{lin}}=\lambda\operatorname{div}u\,I+\mu(Du+Du^T)$, giving the familiar <Navier-Cauchy equation> $-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=f$.
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