= Incompressible limit of isotropic linear elasticity
{title2=$-\mu\Delta u+\nabla p=f,\quad\operatorname{div}u=0$}
Fix a positive <shear modulus> $\mu$ and let the other <Lamé parameter> tend to infinity in $S_\lambda=\lambda\operatorname{tr}(e_\lambda)I+2\mu e_\lambda$. If <strain> and <stress> converge to finite limits and differentiation commutes with the limit, taking the <trace> shows $\operatorname{tr}e_\lambda=O(\lambda^{-1})$. Thus $\operatorname{div}u=0$ and $S=-pI+2\mu e$ for $p=-\lim\lambda\operatorname{tr}e_\lambda$. Force balance becomes the displayed <Stokes equation>, with an elastic <shear modulus> in place of fluid <dynamic viscosity>. The vanishing <volumetric strain> is the linearization of <elastic incompressibility> $\det Dy=1$.
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