The equilateral triangle uniquely minimizes the scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter, settling the Laugesen-Siudeja conjecture and yielding an optimal two-term lower bound.
Shape optimization for the laplacian eigenvalue over triangles and its application to interpolation error analysis.Journal of Differential Equations, 376:750–772
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Sharp Dirichlet eigenvalue inequalities on triangles
The equilateral triangle uniquely minimizes the scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter, settling the Laugesen-Siudeja conjecture and yielding an optimal two-term lower bound.