A note on the Faber-Krahn inequality

  • Tilak Bhattacharya


In this work we study the well known Faber-Krahn  inequality for planar domains. Let u>0 be the first eigenfunction of the Laplacian on a bounded domain and λ_1 be the first eigenvalue. Let λ^∗_1  be the first eigenvalue for the symmetrized domain. We prove that a certain weighted L^1 integral of the isoperimetric deficiencies of the level sets of u may be bounded by the quantity λ_1 − λ^∗_1 . This leads to a sharper version of the Faber-Krahn inequality. It can be easily shown that this result also holds for more general divergence type equations.