On the \(N\)-Cheeger problem for component-wise increasing norms (Q6590490)

The classical Cheeger constant of a measurable set \(\Omega \subset \mathbb{R}^d\) is defined by minimizing the isoperimetric ratio \N\[\Nh(\Omega) := \left\{\frac{\mbox{Per}(E)}{|E|} \ : \ E \subset \Omega, \ |E| >0\right\} \N\]\Nwhere \(\mbox{Per}(E)\) is the distributional perimeter and \(|E|\) is the \(d\)-dimensional Lebesgue measure of \(E.\) It is known in the literature that the geometric quantity \(h(\Omega)\) shares remarkable spectral properties: \N\[\N\left(\frac{h(\Omega)}{p}\right)^p \leq \lambda_{1,p}(\Omega) \ \ \ \mbox{and} \ \ \ h(\Omega) = \lim_{p \rightarrow 1+} \lambda_{1,p}(\Omega),\tag{a}\N\]\Nwhere \( \lambda_{1,p}(\Omega)\) is the first eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega.\)\N\NIn this article, the authors study a natural extension that finds a cluster of \(\Omega\) that minimizes a combination of their isoperimetric ratios. Given the number of partition \(N \in \mathbb{N}\) and a reference function \(\Phi: \mathbb{R}^N \to [0,\infty),\) consider the problem \N\[\NH^{\Phi, N}(\Omega) := \inf \ \Phi \left(\frac{\mbox{Per}(E_1)}{|E_1|} , \cdots, \frac{\mbox{Per}(E_N)}{|E_N|} \right) \N\]\Nwhere the infimum is taken over an appropriate partitions \(E = (E_1, \ldots, E_N)\) of \(\Omega,\) called \textit{\(N\)-clusters} of \(\Omega.\) On the spectral side, there are two natural formulations possible:\N\N\(\bullet\) \(\mathcal{L}^{\Phi,N}_{1,p}(\Omega) := \inf \ \Phi(\lambda_{1,p}(E_1), \cdots, \lambda_{1,p}(E_N)),\) where the infimium is taken over appropriate partitions, called \textit{\(N\)-sets} of \(\Omega,\) and\N\N\(\bullet\) \(\Lambda^{\Phi,N}_{1,p}(\Omega) := \inf \ \Phi (\|\nabla u_1\|_p^p, \cdots, \|\nabla u_N\|_p^p)\), where the infimum is taken over all \(u = (u_1, \ldots, u_N)\) of class \(W_0^{1,p}(\Omega)\) (or, \(BV_0(\Omega)\) if \(p = 1\)) constrained by unitary \(p\)-norm.\N\NThe particular cases of \(\Phi(x) = \|x\|_{\infty}\) and \(\|x\|_1\) has been studied in [\textit{V. Bobkov} and \textit{E. Parini}, J. Lond. Math. Soc., II. Ser. 97, No. 3, 575--600 (2018; Zbl 1394.49041)] and in [\textit{M. Caroccia} and \textit{S. Littig}, J. Convex Anal. 26, No. 1, 33--47 (2019; Zbl 1417.49060)] respectively.\N\NUnder fairly relaxed assumptions on \(\Phi\), such as lower semicontinuity (sometimes continuity), coercivity, and monotonicity (sometimes strict monotonicity), the authors demonstrate the existence and regularity of minimizers. They also establish relationships among the problems above and the stability of the Cheeger constants and their minimizers. In particular, the authors prove\N\N\begin{itemize}\N\item Analogous results of (a) hold for the partition problem.\N\N\item The Cheeger constants \(H^{\Phi, N}(\Omega), \mathcal{L}^{\Phi,N}_{1,p}(\Omega), \Lambda^{\Phi,N}_{1,p}(\Omega)\) and the corresponding minimizers are stable under the \(\Gamma\)-limit of the referrence function: \(\Phi = \Gamma - \lim_{k\rightarrow \infty} \Phi_k.\)\N\end{itemize}\N\NThe article is well-organized and self-contained. In this context, the recent article [J. Lond. Math. Soc., II. Ser. 109, No. 1, Article ID e12840, 55 p. (2024; Zbl 1536.49036)] of the authors together with \textit{V. Franceschi}, and \textit{A. Pinamonti}, in the abstract setting would also be a good accessory read for interested readers.