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About Bounds for Eigenvalues of the Laplacian with Density
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in\mathbb{N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and $\mathrm{d}v_g$ respectively, the Riemannian metric on $M$ and the associated volume element. Let $\Delta$ be the Laplace operator on $M$ equipped with the weighted volume form $\mathrm{d}m:= \mathrm{e}^{-h}\,\mathrm{d}v_g$. We are interested in the operator $L_h\cdot:=\mathrm{e}^{-h(\alpha-1)}(\Delta\cdot +\alpha g(\nabla h,\nabla\cdot))$, where $\alpha > 1$ and $h\in C^2(M)$ are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian $L_h$ with the Neumann boundary condition if the boundary is non-empty.
Keywords: eigenvalue, Laplacian, density, Cheeger inequality, upper bounds. 
@article{SIGMA_2020_16_a89,
     author = {A{\"\i}ssatou Moss\`ele Ndiaye},
     title = {About {Bounds} for {Eigenvalues} of the {Laplacian} with {Density}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a89/}
}
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Aïssatou Mossèle Ndiaye. About Bounds for Eigenvalues of the Laplacian with Density. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a89/

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