Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates
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2021
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| author | Brandolini, Barbara Chiacchio, Francesco Langford, Jeffrey J. |
| author_facet | Brandolini, Barbara Chiacchio, Francesco Langford, Jeffrey J. |
| contents | We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $γ$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of points that lie on one side of $γ$ and within a prescribed distance $δ(s)$ from $γ(s)$ (here $s$ denotes the arc length parameter for $γ$). Write $μ_1^{odd}(D)$ for the lowest nonzero eigenvalue of the Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the $y$-axis. For all $p>1$, we provide a lower bound on $μ_1^{odd}(D)$ when the distance function $δ$ and the signed curvature $k$ of $γ$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $μ_1^{odd}(D)=μ_1(D)$. We finally study the asymptotics of $μ_1(D)$ as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann $p$-Laplace problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_13903 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates Brandolini, Barbara Chiacchio, Francesco Langford, Jeffrey J. Analysis of PDEs 35P30, 35J92, 35P15 We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $γ$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of points that lie on one side of $γ$ and within a prescribed distance $δ(s)$ from $γ(s)$ (here $s$ denotes the arc length parameter for $γ$). Write $μ_1^{odd}(D)$ for the lowest nonzero eigenvalue of the Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the $y$-axis. For all $p>1$, we provide a lower bound on $μ_1^{odd}(D)$ when the distance function $δ$ and the signed curvature $k$ of $γ$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $μ_1^{odd}(D)=μ_1(D)$. We finally study the asymptotics of $μ_1(D)$ as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann $p$-Laplace problem. |
| title | Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates |
| topic | Analysis of PDEs 35P30, 35J92, 35P15 |
| url | https://arxiv.org/abs/2106.13903 |