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arXiv · 2605.20488

Essential spectrum for the $p-$Laplacian

Lorenzo Brasco·Luca Briani·Giovanni Franzina
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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 $p=2$.

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Lorenzo Brasco, Luca Briani, Giovanni Franzina. 2026-05-19. Essential spectrum for the $p-$Laplacian. https://arxiv.org/abs/2605.20488

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