Eigenvalues of elliptic operators with density

Colbois, Bruno; Provenzano, Luigi

doi:10.1007/s00526-018-1307-0

Cited by 14 publications

(12 citation statements)

References 32 publications

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“…We mention that a similar behavior has been observed for upper bounds on the Neumann eigenvalues of linear elliptic operators of order 2m, m ∈ N and density on Euclidean domains, see [13]. In particular, if 2 ≤ 2m ≤ n uniform upper bounds hold, which morally correspond to the conformal upper bounds discussed in this paper.…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 79%

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

Colbois,

Provenzano

2020

Preprint

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In this note we present upper bounds for the variational eigenvalues of the p-Laplacian on smooth domains of complete n-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold (M, g) for 1 < p ≤ n, and upper bounds for all p > 1 when we fix a metric g. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

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Section: Introduction and Statement Of The Main Resultssupporting

confidence: 79%

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

Colbois,

Provenzano

2020

Preprint

Self Cite

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show abstract

“…We mention that a behavior similar to that highlighted in Theorems 1.1-1.4 has been observed for upper bounds on the Neumann eigenvalues of linear elliptic operators of order 2m, m ∈ N and density on Euclidean domains; see [13]. In particular, if 2 2m n uniform upper bounds hold, which morally correspond to the conformal upper bounds discussed in this paper.…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 77%

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Colbois

Provenzano

2021

Journal of London Math Soc

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In this note we present upper bounds for the variational eigenvalues of the p-Laplacian on smooth domains of complete n-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given metric for 1 < p n, and upper bounds for all p > 1 when we fix a metric. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

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“…Finally, we mention that a behavior similar to that of our case p > n has been observed for upper bounds on the Neumann eigenvalues of linear elliptic operators of order 2m, m ∈ N and density on Euclidean domains (see [8]) and for upper bounds on Neumann eigenvalues of the p-Laplacian in the conformal class of a given metric in a complete Riemannian manifold (see [9]).…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 72%

Upper bounds for the Steklov eigenvalues of the $p$-Laplacian

Provenzano¹

2021

Preprint

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In this note we present upper bounds for the variational eigenvalues of the Steklov p-Laplacian on domains of R n , n ≥ 2. We show that for 1 < p ≤ n the variational eigenvalues σ p,k are bounded above in terms of k, p, n and |∂Ω| only. In the case p > n upper bounds depend on a geometric constant D(Ω), the (n − 1)-distortion of Ω which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for p > n and that the corresponding inequality is sharp, providing examples of domains with boundary measure uniformly bounded away from zero and infinity and arbitrarily large variational eigenvalues.

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Eigenvalues of elliptic operators with density

Cited by 14 publications

References 32 publications

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Upper bounds for the Steklov eigenvalues of the $p$-Laplacian

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