Let $T$ be the interior of a triangle of side length $a,b,c$ and interior angles $\alpha,\beta,\gamma$. Can the smallest (non-zero) eigenvalue of $-\Delta u=\lambda u$ in $T$ and $u=0$ on $\partial T$, be expressed in terms of the side lengths and interior angles?
2026-03-29 14:57:44.1774796264
Principal eigenvalue of Dirichlet-Laplacian on a triangle
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It can be expressed purely in terms of the side lengths, since the smallest nonzero eigenvalue correspond to the optimal Poincare constant; see https://en.wikipedia.org/wiki/Poincar%C3%A9_inequality#The_Poincar.C3.A9_constant
It has been proven in the case that the domain is convex (which a triangle is), that the constant is given in terms of the diameter, which depends upon the side lengths.