Neumann Eigenvalue Sums on Triangles Are (mostly) Minimal for Equilaterals
نویسندگان
چکیده
We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle. Mathematics subject classification (2010): Primary 35P15. Secondary 35J20.
منابع مشابه
Minimizing Neumann Fundamental Tones of Triangles: an Optimal Poincaré Inequality
The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than π/3. Antisymmetry is proved for apertures greater than π/3.
متن کاملMaximizing Neumann Fundamental Tones of Triangles
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.
متن کاملSums of Magnetic Eigenvalues Are Maximal on Rotationally Symmetric Domains
The sum of the first n ≥ 1 energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area) on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The...
متن کاملComputing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice
An efficient semi-analytic method is developed for computing the band structures of two-dimensional photonic crystals which are triangular lattices of circular cylinders. The problem is formulated as an eigenvalue problem for a given frequency using the Dirichlet-to-Neumann (DtN) map of a hexagon unit cell. This is a linear eigenvalue problem even if the material is dispersive, where the eigenv...
متن کاملSums of Laplace Eigenvalues — Rotationally Symmetric Maximizers in the Plane
The sum of the first n ≥ 1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio (area)/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundar...
متن کامل