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This article is cited in 1 scientific paper (total in 1 paper)
About Bounds for Eigenvalues of the Laplacian with Density
Aïssatou Mossèle Ndiaye Institut de Mathématiques, Université de Neuchâtel, Switzerland
Abstract:
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.
Received: February 13, 2020; in final form September 1, 2020; Published online September 25, 2020
Citation:
Aïssatou Mossèle Ndiaye, “About Bounds for Eigenvalues of the Laplacian with Density”, SIGMA, 16 (2020), 090, 8 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1627 https://www.mathnet.ru/eng/sigma/v16/p90
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