Faber–Krahn for polyharmonic operators

Establish whether Euclidean balls minimize the first Dirichlet eigenvalue of the polyharmonic operator (−Δ)^m among all open subsets of R^d with a fixed Lebesgue measure, for general dimensions d≥2 and orders m≥1 (beyond the biharmonic cases known in d=2 and d=3).

Background

For the Dirichlet Laplacian, the Rayleigh–Faber–Krahn theorem asserts that balls minimize the first eigenvalue among sets of fixed volume. An analogous minimization principle for higher-order operators is far less understood.

The paper highlights that, unlike the classical Laplacian, the question for the polyharmonic operator (−Δ)^m remains unresolved in general, with only special results available for the biharmonic case (m=2) in two and three dimensions.