Validity of Ivrii’s dynamical condition for smooth domains

Determine whether Ivrii’s dynamical condition is satisfied for all bounded open sets in R^d with smooth boundary, thereby establishing the two-term Weyl asymptotic expansion for the eigenvalue counting function of the Dirichlet and Neumann Laplacians on every smooth domain.

Background

Ivrii proved two-term asymptotics for the eigenvalue counting function under an additional dynamical condition on the domain, beyond boundary smoothness. The authors emphasize that this dynamical condition is believed to hold for all smooth domains, which would make the two-term Weyl expansion universally valid in that class, but this belief has not been confirmed in general.

They highlight that verification of the dynamical condition has been achieved only in very limited cases, so resolving its validity for all smooth domains would settle a longstanding question linked to spectral asymptotics.

References

We do not recall the definition of Ivrii’s dynamical condition, as we will not need it in what follows, but we note that it is believed to be satisfied for all open sets Ω with smooth boundary. This conjecture has been verified in a very limited number of cases, but remains open in general.

— Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains (2407.11808 - Frank et al., 16 Jul 2024) in Section 1 (Introduction and main results), discussion around formula (4)