Maximizing Neumann fundamental tones of triangles
Richard Snyder Laugesen, Bartłomiej A. Siudeja · Journal of Mathematical Physics · 2009
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 proven for the harmonic and arithmetic means of the first two nonzero eigenvalues.