Hadamard type variation formulas for the eigenvalues of the $η$-Laplacian and applications
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866914215300694016 |
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| author | Gomes, J. N. V. Marrocos, M. A. M. Mesquita, R. R. |
| author_facet | Gomes, J. N. V. Marrocos, M. A. M. Mesquita, R. R. |
| contents | We consider an analytic family of Riemannian metrics on a compact smooth manifold $M$. We assume the Dirichlet boundary condition for the $η$-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all $C^r$ Riemannian metrics $\mathcal{M}^r$ on $M$, all eigenvalues of the $η$-Laplacian are generically simple, for $2\leq r< \infty$. This implies the existence of a residual set of metrics in $\mathcal{M}^r$, which makes the spectrum of the $η$-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions $η$ in the space $\mathcal{F}^r$ of all $C^r$ functions on $M$, which makes again the spectrum of the $η$-Laplacian simple, for $2\leq r< \infty$. Besides, we give a precise information about the complementary of these residual sets, as well as about the structure of the set of deformations of a Riemannian metric (respectively of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard type formulas for the eigenvalues of the $η$-Laplacian in this case. We also establish generic properties of eigenvalues in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1510_07076 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Hadamard type variation formulas for the eigenvalues of the $η$-Laplacian and applications Gomes, J. N. V. Marrocos, M. A. M. Mesquita, R. R. Differential Geometry We consider an analytic family of Riemannian metrics on a compact smooth manifold $M$. We assume the Dirichlet boundary condition for the $η$-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all $C^r$ Riemannian metrics $\mathcal{M}^r$ on $M$, all eigenvalues of the $η$-Laplacian are generically simple, for $2\leq r< \infty$. This implies the existence of a residual set of metrics in $\mathcal{M}^r$, which makes the spectrum of the $η$-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions $η$ in the space $\mathcal{F}^r$ of all $C^r$ functions on $M$, which makes again the spectrum of the $η$-Laplacian simple, for $2\leq r< \infty$. Besides, we give a precise information about the complementary of these residual sets, as well as about the structure of the set of deformations of a Riemannian metric (respectively of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard type formulas for the eigenvalues of the $η$-Laplacian in this case. We also establish generic properties of eigenvalues in this context. |
| title | Hadamard type variation formulas for the eigenvalues of the $η$-Laplacian and applications |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1510.07076 |