Maximizing Neumann fundamental tones of triangles

Abstract

Abstract. We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains.The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.

On the inverse spectral problem for Euclidean triangles

“…In the particular case of the present work, we answer a question raised in Laugesen & Siudeja (2009) by providing what we believe to be compelling computational evidence to the fact that, on the one hand, triangles are spectrally determined by their first three Dirichlet eigenvalues and, on the other hand, this is a peculiarity of this spectral triplet in the sense that for other triplets there will exist pairs of triangles having the corresponding eigenvalues in common.…”

“…a triangle is possible from a finite number of eigenvalues, this number being determined by the first two eigenvalues. Related to this, and in a more recent paper, Laugesen & Siudeja (2009) raised the question of knowing whether the first three Dirichlet eigenvalues l 1 , l 2 and l 3 would uniquely determine (or not) a triangle.…”

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

“…In both plots of figures 14 and 15, we observe that, in each case, m 1 is maximized (among polygons with a given number of sides) by the regular polygon. This result was proved in the case of triangles in Laugesen & Siudeja (2009b) but to our knowledge it is an open problem for the other classes of polygons. It is a similar conjecture to Polya's problem for the Dirichlet case as mentioned in Henrot (2006, § 3.3.3).…”

Dirichlet eigenvalue sums on triangles are minimal for equilaterals

“…We have not succeeded with this approach, unfortunately. We investigated Neumann eigenvalues (rather than Dirichlet) in two recent works [20,21]. The first of those papers maximized the low Neumann eigenvalues of triangles, under perimeter or area normalization, with the maximizer being equilateral.…”