Extend the effectless cut method to higher dimensions

Establish an extension to dimensions n >= 3 of Hersch's effectless cut strategy for first Laplacian eigenvalues on doubly connected domains with Dirichlet boundary conditions on both boundary components: given an open bounded domain Omega = Omega_out \ overline{Omega}_in in R^n, construct an intermediate set G such that Omega_in is compactly contained in G and G is compactly contained in Omega_out, and such that lambda^DD(Omega) = lambda^DN(G cap Omega) = lambda^ND(Omega \ overline{G}), thereby enabling the planar method to generalize beyond two dimensions.

Background

In the plane, Weinberger showed a property that allows one to decompose a doubly connected domain so that the first Dirichlet–Dirichlet eigenvalue equals corresponding mixed (Dirichlet–Neumann and Neumann–Dirichlet) eigenvalues on two subdomains. Hersch used this effectless cut to reduce the problem to isoperimetric inequalities for mixed eigenvalues, thereby obtaining sharp bounds.

The paper notes that directly extending this planar strategy to higher dimensions is delicate because the classical planar tools do not generalize. The authors address this by restricting to axisymmetric domains and proving an effectless cut in that class, but they explicitly state that the general higher-dimensional extension remains open.