Sharp bounds for eigenvalues of triangles

2006/03/27 by Siudeja, B.
#35P15 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/0603630

Abstract

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by π² L²\over 9A², where L is the perimeter and A is the area of this triangle. We show that the constant 9 is optimal and that the optimal constant for the lower bound of the same form is 16. This gives a positive answer to a conjecture made by P. Freitas.

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• Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals
• Dirichlet eigenvalue sums on triangles are minimal for equilaterals
• The second Dirichlet eigenvalue is simple on every non-equilateral triangle
• The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles