= Dirichlet resonance in an equilateral triangle
{c}
{title2=$\lambda=-4\pi^2/(3\ell^2)$}
If $\beta_1,\beta_2,\beta_3$ are the affine barycentric coordinates of an <equilateral triangle> of side $\ell$, then $h=\prod_j\sin(\pi\beta_j)=\frac14\sum_j\sin(2\pi\beta_j)$ vanishes on the boundary and satisfies $\Delta h=-16\pi^2h/(3\ell^2)$. Thus the problem $\Delta q-4\lambda q=0$ with zero <Dirichlet boundary data> has distinct solutions zero and $h$ at the displayed negative parameter. Their normal traces differ, demonstrating failure of a uniquely determined <Dirichlet-to-Neumann map> at resonance.
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