Which 2D domain with fixed area has the lowest laplacian eigenvalue?

Question

Which 2D domain with fixed area has the lowest laplacian eigenvalue?

I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?

Answer 1

This statement (for the Dirichlet boundary condition) is known as the Rayleigh–Faber–Krahn inequality, proved independently by Faber and Krahn about 30 years after Lord Rayleigh conjectured it. So you shouldn't expect to just sit down and prove it on your own.

The EoM article is quite detailed. I'll summarize the main points of the proof:

1. The lowest eigenvalue is the infimum of the ratio $\int |\nabla u|^2\big/\int u^2$ over $u\in H_0^1(\Omega)$.
2. The symmetric decreasing rearrangement of $u$ preserves $L^2$ norm (this is easy).
3. The symmetric decreasing rearrangement of $u$ does not increase the $L^2$ norm of the gradient (this is hard, and is the main point of the proof).
4. Combining 1-3, the statement follows since for any eligible function on $\Omega$ there is another one on the disk whose ratio $\int |\nabla u|^2\big/\int u^2$ is not greater.

Standard references:

• C. Bandle, "Isoperimetric inequalities and applications"
• G. Pólya, G. Szegő, "Isoperimetric inequalities in mathematical physics"