Barta Theorem for the $p$-Laplacian and Geometric Applications
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915861108883456 |
|---|---|
| author | Silva, Paulo Henryque C. |
| author_facet | Silva, Paulo Henryque C. |
| contents | In this article, we develop a Barta-type formulation for the $p$-Laplacian on Riemannian manifolds, extending the approach of Cheung-Leung and Bessa-Montenegro from the linear to the nonlinear setting. This framework yields sharp lower bounds for the $p$-fundamental tone without any assumptions on boundary regularity. As applications, we obtain nonlinear extensions of Cheng's eigenvalue comparison theorem and the Cheng-Li-Yau estimate for $p \geq 2$ in the context of minimal immersions. In particular, under the above assumptions, the domain $Ω$ is $p$-stable for the Schrödinger-type operator associated with the potential $\mathcal{V} = \|A\|^{p}$, where $A$ denotes the second fundamental form of the minimal immersion. In addition, we establish a lower bound for the $p$-fundamental tone in the setting where the immersion has locally bounded mean curvature. Finally, we provide a Kazdan-Kramer type characterization of the $p$-fundamental tone, offering a unified and geometric perspective on spectral bounds for the operator $p$-Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Barta Theorem for the $p$-Laplacian and Geometric Applications Silva, Paulo Henryque C. Analysis of PDEs Differential Geometry Spectral Theory In this article, we develop a Barta-type formulation for the $p$-Laplacian on Riemannian manifolds, extending the approach of Cheung-Leung and Bessa-Montenegro from the linear to the nonlinear setting. This framework yields sharp lower bounds for the $p$-fundamental tone without any assumptions on boundary regularity. As applications, we obtain nonlinear extensions of Cheng's eigenvalue comparison theorem and the Cheng-Li-Yau estimate for $p \geq 2$ in the context of minimal immersions. In particular, under the above assumptions, the domain $Ω$ is $p$-stable for the Schrödinger-type operator associated with the potential $\mathcal{V} = \|A\|^{p}$, where $A$ denotes the second fundamental form of the minimal immersion. In addition, we establish a lower bound for the $p$-fundamental tone in the setting where the immersion has locally bounded mean curvature. Finally, we provide a Kazdan-Kramer type characterization of the $p$-fundamental tone, offering a unified and geometric perspective on spectral bounds for the operator $p$-Laplacian. |
| title | Barta Theorem for the $p$-Laplacian and Geometric Applications |
| topic | Analysis of PDEs Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2603.08350 |