= Solution
For the <displacement gradient tensor> specified in the question,
$$
\nabla\mathbf u:\nabla\mathbf u^T
=\operatorname{tr}(\nabla\mathbf u\,\nabla\mathbf u^T)
=u_x^2+u_y^2+v_x^2+v_y^2,
$$
while the <divergence> is $\nabla\cdot\mathbf u=u_x+v_y$. Hence the <isotropic linear-elastic energy density> is
$$
\frac\mu2(u_x^2+u_y^2+v_x^2+v_y^2)
+\frac{\lambda+\mu}{2}(u_x+v_y)^2.
$$
Expanding the square and collecting terms yields
$$
\boxed{
\mathcal J=\iint_\Omega\left[
\left(\frac\lambda2+\mu\right)(u_x^2+v_y^2)
+\frac\mu2(u_y^2+v_x^2)
+(\lambda+\mu)u_xv_y
\right]dx\,dy}.
$$
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