Relax boundary smoothness assumptions for Robin resolvent asymptotics

Determine how far the boundary smoothness assumptions in the eigenvalue asymptotics for the resolvent difference of Robin operators can be relaxed below smooth (C∞) boundaries while still establishing Weyl-type asymptotic formulas. Specifically, ascertain whether the pseudodifferential approach of Grubb (2014) or the approximation approach developed in Rozenblum (2023) can yield these asymptotics under reduced boundary regularity (e.g., C1,1 or Lipschitz boundaries).

Background

The paper proves Weyl-type asymptotic formulas for the eigenvalues of the resolvent difference in Robin-type problems when the boundary is smooth. This restriction stems from using an approximation scheme where asymptotics for more regular problems are known. For interior perturbations (δ-potentials on Lipschitz hypersurfaces), asymptotics can be derived by approximating with results established for Lipschitz surfaces, but for Robin problems only smooth-boundary results are currently available.

The author notes that existing methods—such as nonsmooth pseudodifferential calculus and approximation techniques for Poincaré–Steklov eigenvalues—might enable relaxing boundary regularity, but the extent to which these can be applied to obtain asymptotics for the Robin resolvent difference remains uncertain.