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

We establish a sharp Faber-Krahn inequality for the normalized (or game-theoretic) $p$-Laplacian, proving that, for $p\ge 2$, 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 $\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }{\mathrm{\Delta }}_p^{N}u$, with $p>1$ and $-1<\gamma \le p-2$. 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.