Fritz Gesztesy (Missouri)
A bound for the eigenvalue counting function for Krein-von Neumann extensions of uniformly elliptic second order PDE operators
We derive a bound for the eigenvalue counting function for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets \Omega in R^n. (No assumptions on the boundary of \Omega are made; the coefficients are supposed to satisfy certain regularity conditions.)
Our technique relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n.
We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension.
This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.