= Dirichlet reaction-diffusion growth threshold in a cuboid
{c}
{title2=$s_{111}=\gamma-\mu\pi^2(L_x^{-2}+L_y^{-2}+L_z^{-2})$}
For $n_t=\mu\Delta n+\gamma n$ in a rectangular box with zero <Dirichlet boundary conditions>, <separation of variables> gives spatial <eigenfunctions> $\sin(p\pi x/L_x)\sin(q\pi y/L_y)\sin(r\pi z/L_z)$ and temporal rates $\gamma-\mu\pi^2(p^2/L_x^2+q^2/L_y^2+r^2/L_z^2)$. The largest rate is the displayed one. Every nonzero nonnegative initial density has positive projection onto this first <eigenfunction>, so its sign determines eventual exponential growth or decay; equality leaves a stationary first mode. At fixed volume, write $a_j=L_j^{-2}$: their product is fixed, and the <arithmetic-geometric mean inequality> minimizes their sum when all three are equal. Thus the cube maximizes the growth rate among rectangular boxes of fixed volume; making a box sufficiently thin suppresses growth.
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