= Solution
Vary each field component independently and integrate the gradient term by parts, taking fixed boundary values or vanishing surface variations. With $\rho=m_1^2+m_2^2+m_3^2$, the <Euler-Lagrange equations> for the free-energy saddle are
$$
\boxed{\begin{aligned}
-\nabla^2m_1+(t+r)m_1+4u\rho m_1&=0,\\
-\nabla^2m_2+(t-r)m_2+4u\rho m_2&=0,\\
-\nabla^2m_3+t m_3+4u\rho m_3&=0.
\end{aligned}}
$$
The anisotropy shifts the first two inverse susceptibilities in opposite directions. The derivative of $u\rho^2$ is $4u\rho m_j$, which fixes the factor needed for both the <magnetization> and the free energies in <3/c>. Under natural free-boundary conditions the integration by parts instead supplies $\partial_n m_j=0$ at the surface.
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