Spectral Properties of the Logarithmic Laplacian with Indefinite Weights
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916009546350592 |
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| author | Arora, Rakesh Mukherjee, Tuhina Vaishnavi, Arshi |
| author_facet | Arora, Rakesh Mukherjee, Tuhina Vaishnavi, Arshi |
| contents | In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13513 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Properties of the Logarithmic Laplacian with Indefinite Weights Arora, Rakesh Mukherjee, Tuhina Vaishnavi, Arshi Analysis of PDEs In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain. |
| title | Spectral Properties of the Logarithmic Laplacian with Indefinite Weights |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.13513 |