On the $N$-Cheeger problem for component-wise increasing norms

We study Cheeger and $p$-eigenvalue partition problems depending on a given evaluation function $\mathrm{\Phi }$ for $p\in [1,\mathrm{\infty })$. We prove existence and regularity of minima, relations among the problems, convergence, and stability with respect to $p$ and to $\mathrm{\Phi }$.