Optimal domains for the Cheeger inequality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908584436039680 |
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| author | Bucur, Dorin Buttazzo, Giuseppe de Villeroché, Alexis |
| author_facet | Bucur, Dorin Buttazzo, Giuseppe de Villeroché, Alexis |
| contents | 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 $1\le q<p\le+\infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal domains for the Cheeger inequality Bucur, Dorin Buttazzo, Giuseppe de Villeroché, Alexis Optimization and Control 49Q10, 49J45, 49R05, 35P15, 35J25 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 $1\le q<p\le+\infty$ 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. |
| title | Optimal domains for the Cheeger inequality |
| topic | Optimization and Control 49Q10, 49J45, 49R05, 35P15, 35J25 |
| url | https://arxiv.org/abs/2510.08032 |