First Dirichlet eigenvalue of the weighted 1-Laplacian operator
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2026
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| author | Barbato, R. de Lis, J. C. Sabina de León, S. Segura |
| author_facet | Barbato, R. de Lis, J. C. Sabina de León, S. Segura |
| contents | In this paper, we study the eigenvalue problem \[\left\{\begin{array}{cl}-\hbox{div}\left(a(x)\frac{Du}{|Du|}\right)=Λ\, b(x)\frac{u}{|u|} & \text{in }Ω\\u=0 & \text{on }\partialΩ,\end{array}\right.\] where $a(x)$ and $b(x)$ are suitable nonnegative functions. We prove that the first eigenvalue coincides with the weighted Cheeger constant. To see this identity, we analyze the behavior of the first Dirichlet eigenvalue of the weighted $p$-Laplacian operator as $p$ goes to $1$. In the case that the weight $a(x)$ is Lipschitz-continuous, we show that the limit of eigenvalues of the weighted $p$-Laplacian exists, and it is the weighted Cheeger constant. In addition, we check that the sequence of normalized $p$-eigenfunctions converges to the normalized eigenfunction of our limiting problem, which turns out to be bounded. For more general weights, we identify the first eigenvalue of the weighted 1-Laplacian operator with the weighted Cheeger constant and prove that the associated eigenfunction is bounded. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | First Dirichlet eigenvalue of the weighted 1-Laplacian operator Barbato, R. de Lis, J. C. Sabina de León, S. Segura Analysis of PDEs 35P30, 35J60, 35J92 In this paper, we study the eigenvalue problem \[\left\{\begin{array}{cl}-\hbox{div}\left(a(x)\frac{Du}{|Du|}\right)=Λ\, b(x)\frac{u}{|u|} & \text{in }Ω\\u=0 & \text{on }\partialΩ,\end{array}\right.\] where $a(x)$ and $b(x)$ are suitable nonnegative functions. We prove that the first eigenvalue coincides with the weighted Cheeger constant. To see this identity, we analyze the behavior of the first Dirichlet eigenvalue of the weighted $p$-Laplacian operator as $p$ goes to $1$. In the case that the weight $a(x)$ is Lipschitz-continuous, we show that the limit of eigenvalues of the weighted $p$-Laplacian exists, and it is the weighted Cheeger constant. In addition, we check that the sequence of normalized $p$-eigenfunctions converges to the normalized eigenfunction of our limiting problem, which turns out to be bounded. For more general weights, we identify the first eigenvalue of the weighted 1-Laplacian operator with the weighted Cheeger constant and prove that the associated eigenfunction is bounded. |
| title | First Dirichlet eigenvalue of the weighted 1-Laplacian operator |
| topic | Analysis of PDEs 35P30, 35J60, 35J92 |
| url | https://arxiv.org/abs/2605.25642 |