Leander Geisinger: Refined semiclassics for Laplace operators on bounded domains

We consider the Laplacian -Δ and the fractional Laplacian (-Δ)^s, 0<s<1, on a domain and investigate the asymptotic behavior of the eigenvalues. Extending methods from semi-classical analysis we prove a two-term formula for the sum of the eigenvalues with the leading (Weyl) term given by the volume and the subleading term depending on properties of the boundary. These results are valid under weak assumptions on the regularity of the boundary. In the local case, s=1, we show how the second term depends on different boundary conditions, including Dirichlet, Neumann and varying Robin conditions. For Dirichlet boundary conditions, we also discuss relations of the asymptotic results to improved uniform estimates for eigenvalue sums and eigenvalue means.