Sharp constant in inradius-based lower bounds for polyharmonic eigenvalues

Ascertain the optimal constant in the lower bound that controls the first Dirichlet eigenvalue of the polyharmonic operator (−Δ)^m on convex domains in R^d in terms of the inradius R_Ω, as given by Owen’s analogue of the Hersch–Protter inequality, for orders m≥2.

Background

For the Laplacian, Hersch–Protter-type bounds yield sharp inradius-based lower estimates. Owen extended such estimates to polyharmonic operators, but the optimal constant in these inequalities is unknown for higher orders.

Determining the sharp constant would clarify the precise geometric dependence of the lowest eigenvalue on the inradius for polyharmonic operators on convex domains.