Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition

Question

Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one?

Answer 1

Yes. Steiner symmetrization decreases the first Dirichlet eigenvalue (also known as the fundamental frequency), unless the domain is already symmetric. And a triangle that is not equilateral can be Steiner-symmetrized in a nontrivial way.

More generally, Pólya and Szegő conjectured that among all $n$-gons of fixed area the regular $n$-gon has the lowest fundamental frequency. This has been proved (by symmetrization) for $n=3,4$. Remains open for $n\ge 5$.

I don't have a reference to original source, but the statement can be found in

• Antoine Henrot. Minimization problems for eigenvalues of the Laplacian. Journal of Evolution Equations, Springer Verlag, 2003, 3, pp.443-461.
• Bartłomiej Siudeja. Isoperimetric inequalities for eigenvalues of triangles, Indiana Univ. Math. J. 59 (2010), no. 3, 1097–1120

Also, the classical source of such problems is

• George Pólya, Gábor Szegő. Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, 1951