The Limit as p → 1 of the Higher Eigenvalues of the p-Laplacian Operator -Δp

This work provides a direct proof of the existence for each n is an element of N of the limit lambda((1),n) := lim(p -> 1) lambda((p),n) of the n-th Ljusternik-Schnirelman Dirichlet eigenvalue lambda((p),n) of -Delta(p) in a bounded Lipschitz domain Omega subset of R-N. Most importantly, it is shown that lambda((1),n) defines an eigenvalue of the 1-Laplacian operator -Delta(1), with a well-defined strong associated eigenfunction u(n) is an element of BV(Omega). In the main results of the paper, the radial LS eigenvalues of -Delta(1) are fully described, together with a detailed account on the profiles of their associated eigenfunctions. Our approach does not involve critical point theory for non-smooth functionals, although the definition of the LS-spectrum of -Delta(1) relies on it.