Essential spectrum for the $p-$Laplacian

TL;DR

This paper introduces a new variational approach to the essential spectrum of the nonlinear Dirichlet p-Laplacian, extending classical results and providing geometric characterizations, with explicit spectrum computations on specific domains.

Contribution

It develops a variational notion of essential spectrum for the p-Laplacian, extending Persson's theorem to nonlinear operators and computing the spectrum on a rectilinear strip.

Findings

The essential spectrum is purely essential with no embedded eigenvalues on a rectilinear strip.

The new construction aligns with classical theory when p=2.

Elementary proofs are provided for the nonlinear setting.

Abstract

We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincar\'e constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.