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PreprintCC BY 4.0arXiv

Spectral geometry of the magnetic p-Laplacian

  • David Krejčiřík
  • Rossano Sannipoli
Open at sourcePDFDOICite

Abstract

We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear p-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.

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Cite

BibTeXRIS
BibTeX
@misc{krejcirik2026spectral,
  title = {{Spectral geometry of the magnetic $p$-Laplacian}},
  author = {David Krejčiřík and Rossano Sannipoli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37313},
  eprint = {2609.37313},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37313},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectral geometry of the magnetic p-Laplacian
AU  - David Krejčiřík
AU  - Rossano Sannipoli
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37313
UR  - https://arxiv.org/abs/2609.37313
AB  - We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear p-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.
ER  -