{"ID":2858063,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.08032","arxiv_id":"2510.08032","title":"Optimal domains for the Cheeger inequality","abstract":"In this paper we consider the scale invariant shape functional $${\\mathcal{F}}_{p,q}(Ω)=\\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\\le q\u003cp\\le+\\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\\partialΩ$. We study both the maximization and minimization problems for ${\\mathcal{F}}_{p,q}$, and show the existence of optimal domains in ${\\mathbb{R}}^d$, along with some of their qualitative properties. Surprisingly, the case of a bounded box $D$ constraint $$\\max\\Big\\{λ_q(Ω)\\ :\\ Ω\\subset D,\\ λ_p(Ω)=1\\Big\\},$$ leads to a problem of different nature, for which the existence of a solution is shown by analyzing optimal capacitary measures. In the last section we list some interesting questions that, in our opinion, deserve to be investigated.","short_abstract":"In this paper we consider the scale invariant shape functional $${\\mathcal{F}}_{p,q}(Ω)=\\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\\le q\u003cp\\le+\\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\\partialΩ$. We stud...","url_abs":"https://arxiv.org/abs/2510.08032","url_pdf":"https://arxiv.org/pdf/2510.08032v1","authors":"[\"Dorin Bucur\",\"Giuseppe Buttazzo\",\"Alexis de Villeroché\"]","published":"2025-10-09T10:10:51Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}