On the Asymptotic Behavior of the First Eigenvalue of Robin Problem With Large Parameter

2015 ◽  
Vol 1 (1) ◽  
pp. 123-135
Author(s):  
Alexey Filinovskiy
Keyword(s):  
Asymptotic Behavior ◽  
First Eigenvalue ◽  
Robin Problem ◽  
Large Parameter ◽  
The First Eigenvalue

scholarly journals WEYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS AND DE GENNES' BOUNDARY CONDITION

2008 ◽  
Vol 20 (08) ◽  
pp. 901-932 ◽  
Author(s):  
AYMAN KACHMAR
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Robin Boundary Condition ◽  
The Self ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Weyl Asymptotics ◽  
Robin Boundary

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10].

scholarly journals On the resonant Lane–Emden problem for the p-Laplacian

2014 ◽  
Vol 16 (04) ◽  
pp. 1350033 ◽  
Author(s):  
Grey Ercole
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Upper Bound ◽  
Ground State ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Explicit Estimate ◽  
Best Constant ◽  
The First Eigenvalue ◽  
First Eigenfunction

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where [Formula: see text].

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS OF THE ROBIN PROBLEM WITH LARGE PARAMETER

2017 ◽  
Vol 22 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalues And Eigenfunctions ◽  
The Difference ◽  
Robin Boundary ◽  
Outward Unit

We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn , n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem for α → +∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions.

scholarly journals The buckling eigenvalue problem in the annulus

2020 ◽  
pp. 2050044
Author(s):  
Davide Buoso ◽  
Enea Parini
Keyword(s):  
Asymptotic Behavior ◽  
Eigenvalue Problem ◽  
Analytical Study ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Domains ◽  
The First Eigenvalue ◽  
Inner Radius

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.

scholarly journals Lichnerowicz-Obata Estimate, Almost Parallel p-form and Almost Product Manifolds

2021 ◽  
Author(s):  
Masayuki Aino
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Hausdorff Approximation

AbstractWe show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of n-dimensional closed Riemannian manifolds with an almost parallel p-form ($$2\le p \le n/2$$ 2 ≤ p ≤ n / 2 ) in $$L^2$$ L 2 -sense, and give a Gromov-Hausdorff approximation to a product $$S^{n-p}\times X$$ S n - p × X under some pinching conditions when $$2\le p<n/2$$ 2 ≤ p < n / 2 .

Sign in / Sign up

Export Citation Format

Share Document