Faber--Krahn inequality for homogeneous normalized p-Laplacian type operators

@misc{crasta2026faberkrahn,
  title = {{Faber--Krahn inequality for homogeneous normalized $p$-Laplacian type operators}},
  author = {Graziano Crasta and Ilaria Fragalà},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23593},
  eprint = {2610.23593},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23593},
}

TY  - UNPB
TI  - Faber--Krahn inequality for homogeneous normalized p-Laplacian type operators
AU  - Graziano Crasta
AU  - Ilaria Fragalà
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23593
UR  - https://arxiv.org/abs/2610.23593
AB  - We establish a sharp Faber-Krahn inequality for the normalized (or game-theoretic) p-Laplacian, proving that, for p≥ 2, the ball uniquely minimizes the principal Dirichlet eigenvalue among bounded connected Lipschitz domains of prescribed volume satisfying a uniform exterior sphere condition. More generally, we prove the same inequality for the family of homogeneous operators |Du|^γΔ_p^N u, with p>1 and −1<γ≤ p−2. Unlike the classical p-Laplacian, these operators lack a variational structure, preventing the use of standard symmetrization arguments. Our proof overcomes this difficulty through a level-set approach: it relies on a flux identity for the eigenvalue, for which a variational interpretation re-emerges in the special case where the domain is a ball.
ER  -