Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian (Q2804145)

In this expository paper, the author presents the historical origins of isoperimetric type inequalities for eigenvalues and describes some modern developments which give sharp upper bounds for sums of eigenvalues of the Laplacian in terms of moment of inertia. This recent progress uses the rotational symmetry and tight frame properties of the rotation orbits. In two dimensions, he proves that the scale-normalized eigenvalue sum \((\lambda_1+\cdots+\lambda_n)\frac{A^3}{I}\) is maximal for the equilateral triangle among all triangles for each \(n\geq 1\), where \(A\) denotes the area and \(I\) is the moment of inertia about the centroid. Some related problems such as the inverse spectral and spectral gap problems for triangular domains are also discussed.NEWLINENEWLINEFor the entire collection see [Zbl 1334.42001].