Reduction of Dimension as a Consequence of Norm‐resolvent Convergence and Applications

Krejčiřı́k, David; Raymond, Nicolas; Royer, Julien; Siegl, Petr

doi:10.1112/s0025579318000013

Cited by 13 publications

(12 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, in recent years there has been an exponential growth of interest in effective models for thin waveguides under various geometric and analytic deformations, see [12,15,17,29,32,9,8,31,30,35,6,34,7] and further references therein. We refer to [27] for a unifying approach to this type of problems.…”

Section: Thin Stripsmentioning

confidence: 99%

Quantum strips in higher dimensions

Krejčiřı́k¹,

Zahradova²

2020

Operators and Matrices

View full text Add to dashboard Cite

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the twodimensional surface by a one-dimensional Schrödinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.

show abstract

Section: Thin Stripsmentioning

confidence: 99%

Quantum strips in higher dimensions

Krejčiřı́k¹,

Zahradova²

2020

Operators and Matrices

View full text Add to dashboard Cite

show abstract

“…The analysis of eigenvalue problems for differential operators on thin domains has attracted noticeable interest in recent years, see e.g., [4,5,6,9,11,12,13,20,23,29,30,36,38,40] and references therein. A somehow complementary point of view is adopted in the asymptotic analysis of domains with small holes or perforations, see e.g., [1,16,19,33,39].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Ferraresso¹,

Provenzano²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In this section we study the asymptotic behavior of µ 1 (D) as the width function goes to 0. For related results when p = 2 and the width is constant, see for instance [10,16,17,18,22] and the references therein.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates

Brandolini¹,

Chiacchio²,

Langford³

2021

Preprint

View full text Add to dashboard Cite

We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if γ denotes a non-closed curve in R 2 symmetric with respect to the y-axis, let D ⊂ R 2 denote the domain of points that lie on one side of γ and within a prescribed distance δ(s) from γ(s) (here s denotes the arc length parameter for γ). Write µ odd 1 (D) for the lowest nonzero eigenvalue of the Neumann p-Laplacian with an eigenfunction that is odd with respect to the y-axis. For all p > 1, we provide a lower bound on µ odd 1 (D) when the distance function δ and the signed curvature k of γ satisfy certain geometric constraints. In the linear case (p = 2), we establish sufficient conditions to guarantee µ odd 1 (D) = µ 1 (D). We finally study the asymptotics of µ 1 (D) as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann p-Laplace problem.

show abstract

Reduction of Dimension as a Consequence of Norm‐resolvent Convergence and Applications

Cited by 13 publications

References 16 publications

Quantum strips in higher dimensions

Quantum strips in higher dimensions

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates

Contact Info

Product

Resources

About