The Hardy-Davies inequality and spectral lower bounds for elliptic operators

In this informal PDE seminar, I introduce the classical Hardy inequality and a generalisation due to E.B. Davies.
Following the results in Davies' influential 1989 monograph Heat kernels and spectral theory, we will see that
the latter inequality can be used to obtain lower bounds for the principal eigenvalue of elliptic operators
acting on \(L^2(\Omega)\) under no regularity assumptions on the domain \(\Omega\).
I will also comment on analogous results for the principal eigenvalue of the p-Laplacian.