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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2605.08818 |
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| _version_ | 1866910204880224256 |
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| author | He, Dangyang |
| author_facet | He, Dangyang |
| contents | Let $n,m\ge 1$, $α\in(0,1)$, and $β\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2α}\nabla_x\bigr)+|x|^{2β}Δ_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if \[ Q=\frac{n+m(β+1-α)}{1-α}, \] then \[ |Ω|^{\frac{Q-1}{Q}}\le C\,P(Ω) \] for every smooth bounded domain $Ω\subset \mathbb R^{n+m}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08818 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isoperimetric Inequality for degenerate elliptic operators of Grushin type He, Dangyang Classical Analysis and ODEs Primary 35J70, Secondary 53C17 Let $n,m\ge 1$, $α\in(0,1)$, and $β\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2α}\nabla_x\bigr)+|x|^{2β}Δ_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if \[ Q=\frac{n+m(β+1-α)}{1-α}, \] then \[ |Ω|^{\frac{Q-1}{Q}}\le C\,P(Ω) \] for every smooth bounded domain $Ω\subset \mathbb R^{n+m}$. |
| title | Isoperimetric Inequality for degenerate elliptic operators of Grushin type |
| topic | Classical Analysis and ODEs Primary 35J70, Secondary 53C17 |
| url | https://arxiv.org/abs/2605.08818 |