First I'm not 100% sure I'm on the good stack for asking my question. I would like to get a bilinear form for linear elasticity that separate a rotational part from a pure divergence part, so starting from the Navier equation \begin{equation} \mu \nabla^2 \mathbf u +(\mu+\lambda)\nabla(\nabla \cdot \mathbf u) + \mathbf f =0 \end{equation}Then I use the vector Laplacian identity $\nabla(\nabla \cdot \mathbf A)= \nabla^2 \mathbf A +\nabla  \times \nabla \times\mathbf A$  to write\begin{equation} (\lambda +2\mu) \nabla^2 \mathbf u +(\mu+\lambda)\nabla  \times (\nabla \times\mathbf u) + \mathbf f =0 \end{equation}So multiplying by a test function $\mathbf v$ I get\begin{equation}\int_\Omega \bigg((\lambda +2\mu) \nabla^2 \mathbf u \cdot \mathbf v +(\mu+\lambda)\nabla  \times (\nabla \times\mathbf u)\cdot \mathbf v  + \mathbf f \cdot \mathbf v  \bigg) d\Omega=0 \end{equation}and using Green formula, \begin{equation}\int_\Omega \bigg((\lambda +2\mu) (\nabla \mathbf u :\nabla \mathbf v) +(\mu+\lambda)(\nabla  \times \mathbf v)\cdot(\nabla \times\mathbf u) + \mathbf f \cdot \mathbf v  \bigg) d\Omega=0 \end{equation}I'm skeptical about this for, since when I compare the bilinear form\begin{equation}a(\mathbf u,\mathbf v) := \int_\Omega \bigg((\lambda +2\mu) (\nabla \mathbf u :\nabla \mathbf v) +(\mu+\lambda)(\nabla  \times \mathbf v)\cdot(\nabla \times\mathbf u) \bigg) d\Omega\end{equation}With the classical one\begin{equation}a(\mathbf u,\mathbf v) := \int_\Omega \sigma(\mathbf u):\varepsilon(\mathbf v) d\Omega\end{equation}I got few different terms. For instance, in 2D the classical form (noted $a_1$) gives \begin{eqnarray}a_1(\mathbf u,\mathbf v) &=& \int_\Omega \sigma(\mathbf u):\varepsilon(\mathbf v) d\Omega \\&=& \bigg(2\mu \frac{\partial u_x}{\partial x}+\lambda(\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}) \bigg)\frac{\partial v_x}{\partial x}+\mu\bigg(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\bigg)\bigg(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\bigg)+\bigg(2\mu \frac{\partial u_y}{\partial y}+\lambda(\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y})\bigg) \frac{\partial v_y}{\partial y}\\&=&(\lambda+2\mu) \bigg(\frac{\partial u_x}{\partial x} \frac{\partial v_x}{\partial x} +\frac{\partial u_y}{\partial y}\frac{\partial v_y}{\partial y}\bigg) + \mu\bigg(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\bigg)\bigg(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\bigg)+\lambda\bigg(\frac{\partial u_y}{\partial y}\frac{\partial v_x}{\partial x} +\frac{\partial u_x}{\partial x}\frac{\partial v_y}{\partial y}\bigg)\end{eqnarray}Whereas, the rotational form gives\begin{eqnarray}a_2(\mathbf u,\mathbf v) &=& \int_\Omega \bigg((\lambda +2\mu) (\nabla \mathbf u :\nabla \mathbf v) +(\mu+\lambda)(\nabla  \times \mathbf v)\cdot(\nabla \times\mathbf u) \bigg) d\Omega\\&=& (2\mu+\lambda)\bigg(\frac{\partial u_x}{\partial x}\frac{\partial v_x}{\partial x} +\frac{\partial u_x}{\partial y}\frac{\partial v_x}{\partial y}+\frac{\partial u_y}{\partial x}\frac{\partial v_y}{\partial x} +\frac{\partial u_y}{\partial y}\frac{\partial v_y}{\partial y}\bigg)+ (\lambda+\mu)\bigg(\frac{\partial u_y}{\partial x} -\frac{\partial u_x}{\partial y}\bigg)\bigg(\frac{\partial v_x}{\partial y} -\frac{\partial v_y}{\partial x}\bigg)\end{eqnarray}The $(\lambda +\mu)$ terms may be developped and some terms may be factorized with some other terms of $(2\mu+\lambda)$\begin{eqnarray}a_2(\mathbf u,\mathbf v) &=& (2\mu+\lambda)\bigg(\frac{\partial u_x}{\partial x}\frac{\partial v_x}{\partial x}  +\frac{\partial u_y}{\partial y}\frac{\partial v_y}{\partial y}\bigg)+\mu\bigg(\frac{\partial u_x}{\partial y}\frac{\partial v_x}{\partial y} + \frac{\partial u_y}{\partial x}\frac{\partial v_y}{\partial x}\bigg)+ (\lambda+\mu)\bigg(\frac{\partial u_y}{\partial x}\frac{\partial v_x}{\partial y} +\frac{\partial u_x}{\partial y}\frac{\partial v_y}{\partial x}\bigg)\end{eqnarray}The $\mu$ terms in two and second expressions may be factorized \begin{eqnarray}a_2(\mathbf u,\mathbf v) &=& (2\mu+\lambda)\bigg(\frac{\partial u_x}{\partial x}\frac{\partial v_x}{\partial x} + \frac{\partial u_y}{\partial y}\frac{\partial v_y}{\partial y}\bigg)+\mu\bigg(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\bigg)\bigg(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\bigg)+ \lambda\bigg(\frac{\partial u_y}{\partial x}\frac{\partial v_x}{\partial y} +\frac{\partial u_x}{\partial y}\frac{\partial v_y}{\partial x}\bigg)\end{eqnarray}This equation is almost the same as $a_1$ except for the $\lambda$ term.Could someone correct me ?PS: the rotational being defined by\begin{eqnarray}\nabla \times \mathbf u =\left|\begin{array}{l}\partial_x\\\partial_y \\\partial_z\end{array}\right.\times\left|\begin{array}{l}u_x\\u_y \\u_z\end{array}\right.=\left|\begin{array}{l}\partial_y u_z -\partial_z u_y\\\partial_z u_x -\partial_x u_z \\\partial_x u_y -\partial_y u_x\end{array}\right.\end{eqnarray}We have\begin{eqnarray}(\nabla \times \nabla \times \mathbf u ) \cdot \mathbf v =\left|\begin{array}{l}\partial_y (\partial_x u_y -\partial_y u_x) - \partial_z(\partial_z u_x -\partial_x u_z)\\\partial_z (\partial_y u_z -\partial_z u_y) - \partial_x(\partial_x u_y -\partial_y u_x)\\\partial_x (\partial_z u_x -\partial_x u_z) - \partial_y (\partial_y u_z -\partial_z u_y)\end{array}\right.\cdot \left|\begin{array}{l}v_x \\v_y\\v_z\end{array}\right.\end{eqnarray}And using integration by part (but I suspect that the flaw stands here), \begin{eqnarray}\int_\Omega (\nabla \times \nabla \times \mathbf u ) \cdot \mathbf v &=&-\int_\Omega \partial_y v_x(\partial_x u_y -\partial_y u_x) - \partial_z v_x (\partial_z u_x -\partial_x u_z)\\&&+\partial_z  v_y(\partial_y u_z -\partial_z u_y) - \partial_x v_y (\partial_x u_y -\partial_y u_x)\\&&+\partial_x v_z(\partial_z u_x -\partial_x u_z) - \partial_y v_z(\partial_y u_z -\partial_z u_y)\\&=& - \int_\Omega (\partial_y v_x - \partial_x v_y )(\partial_x u_y -\partial_y u_x) +(\partial_z  v_y-\partial_y v_z) (\partial_y u_z -\partial_z u_y) +(\partial_x v_z- \partial_z v_x )(\partial_z u_x -\partial_x u_z)\\&=& \int_\Omega (\nabla \times \mathbf v )\cdot (\nabla \times \mathbf u)\end{eqnarray}Thank you very much.
variational formulation of linear elasticity
pde;well posedness;elliptic pde