On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| author | Bessa, Gregório Pacelli F. Jorge, Luquésio Petrola de M. Mari, Luciano |
| author_facet | Bessa, Gregório Pacelli F. Jorge, Luquésio Petrola de M. Mari, Luciano |
| contents | In this paper, we study the principal eigenvalue $μ(\mathscr{F}_k^-,E)$ of the fully nonlinear operator
\[
\mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u|
\] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and $\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $\nabla^2 u$. We prove a lower estimate for $μ(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_Ψ(E)$, for suitable $Ψ$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate
\[
μ(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}.
\] The theorem is used to study the spectrum of bounded submanifolds in $\mathbb{R}^n$, improving on our previous work in the direction of a question posed by S.T. Yau. In particular, the result applies to solutions of Plateau's problem for CMC surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_14740 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature Bessa, Gregório Pacelli F. Jorge, Luquésio Petrola de M. Mari, Luciano Analysis of PDEs Differential Geometry Spectral Theory In this paper, we study the principal eigenvalue $μ(\mathscr{F}_k^-,E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and $\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $\nabla^2 u$. We prove a lower estimate for $μ(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_Ψ(E)$, for suitable $Ψ$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate \[ μ(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}. \] The theorem is used to study the spectrum of bounded submanifolds in $\mathbb{R}^n$, improving on our previous work in the direction of a question posed by S.T. Yau. In particular, the result applies to solutions of Plateau's problem for CMC surfaces. |
| title | On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature |
| topic | Analysis of PDEs Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2109.14740 |