Essential spectrum for the $p-$Laplacian
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918513048813568 |
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| author | Brasco, Lorenzo Briani, Luca Franzina, Giovanni |
| author_facet | Brasco, Lorenzo Briani, Luca Franzina, Giovanni |
| contents | 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 $L^p$ Poincaré 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 $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20488 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Essential spectrum for the $p-$Laplacian Brasco, Lorenzo Briani, Luca Franzina, Giovanni Analysis of PDEs Spectral Theory 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 $L^p$ Poincaré 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 $p=2$. |
| title | Essential spectrum for the $p-$Laplacian |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2605.20488 |