Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912776211922944 |
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| author | Langharst, Dylan Roysdon, Michael |
| author_facet | Langharst, Dylan Roysdon, Michael |
| contents | Recently, Haddad, Jiménez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator $Δ_{Q,p}^\mathcal{A} f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval.
Given $n,m \in \mathbb{N}$, a sufficiently smooth convex body $Q \subset \mathbb{R}^m$, a bounded, open set $Ω\subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} Δ_{Q,p}^\mathcal{A} f = λ_{1,p}^\mathcal{A}(Q,Ω) |f|^{p-2} f &\text{ in } Ω; \\ f=0 & \text{ on } \partial Ω, \end{cases} \] for $f \in W^{1,p}_0(Ω)$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator Langharst, Dylan Roysdon, Michael Functional Analysis Analysis of PDEs Metric Geometry Primary: 35P30, 35B09 Secondary: 52A20 Recently, Haddad, Jiménez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator $Δ_{Q,p}^\mathcal{A} f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval. Given $n,m \in \mathbb{N}$, a sufficiently smooth convex body $Q \subset \mathbb{R}^m$, a bounded, open set $Ω\subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} Δ_{Q,p}^\mathcal{A} f = λ_{1,p}^\mathcal{A}(Q,Ω) |f|^{p-2} f &\text{ in } Ω; \\ f=0 & \text{ on } \partial Ω, \end{cases} \] for $f \in W^{1,p}_0(Ω)$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities. |
| title | Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator |
| topic | Functional Analysis Analysis of PDEs Metric Geometry Primary: 35P30, 35B09 Secondary: 52A20 |
| url | https://arxiv.org/abs/2512.17237 |