On the asymptotic behavior of the principal eigenvalues of some elliptic problems

Godoy, T.; Gossez, Jean–Pierre; Paczka, S.

doi:10.1007/s10231-009-0120-y

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…for all ϕ ∈ S λ per+ . Following the same arguments in Lemma 2.1, we can conclude from the definition (19)…”

Section: Shuang Liu and Yuan Loumentioning

confidence: 82%

“…The proof of Theorem 1.1 was first given by Godoy et al [19] via a variant of the min-max formula derived in [18] for principal eigenvalues. Our proof relies heavily on properties of functional J A defined in (2) by identifying the definition cone S as…”

mentioning

confidence: 99%

“…We thank the referee for helpful suggestions. We thank Professor Nadin for bringing Reference [19] to our attention. We thank Professors Berestycki and Hamel for their helpful discussions.…”

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confidence: 99%

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A functional approach towards eigenvalue problems associated with incompressible flow

Liu¹,

Lou²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator L A = −div(a(x)∇) + AV · ∇ + c(x) and its adjoint operator for general incompressible flow V. The functional can be applied to establish the monotonicity of the principal eigenvalue λ 1 (A), as a function of the advection amplitude A, for the operator L A subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed c * (A)/A for general incompressible flow, where c * (A) is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.which model various physical, chemical, and biological processes: On unbounded domains [17,44], compact manifolds [10], and bounded domains with appropriate boundary conditions [1,5,7,34]. Function w represents the density of a population or a substance diffusing with diffusion matrix a(x), reacting through the nonlinearity wf , and advected by the stationary fluid flow V in heterogeneous media.Let Ω be a bounded region of R N with smooth boundary ∂Ω, and n(x) be the outward unit normal vector at x ∈ ∂Ω. Consider equation (1) defined on Ω and suppose that w satisfies bw +(1−b)[a(x)∇w]·n = 0 on ∂Ω with parameter b ∈ [0, 1]. The stability of steady state w ≡ 0 and the minimal speed c * (A) of traveling fronts for equation (1) are associated with the principal eigenvalue, denoted as λ 1 (A), for the linear eigenvalue problem L A u := −div(a(x)∇u) + AV · ∇u + c(x)u = λ 1 (A)u, subject to appropriate boundary conditions, where c(x) = −f (x, 0). 2010 Mathematics Subject Classification. Primary: 35J20, 35J60, 35P15.

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“…for all ϕ ∈ S λ per+ . Following the same arguments in Lemma 2.1, we can conclude from the definition (19)…”

Section: Shuang Liu and Yuan Loumentioning

confidence: 82%

mentioning

confidence: 99%

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A functional approach towards eigenvalue problems associated with incompressible flow

Liu¹,

Lou²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…There exists an enormous body of literature for these kinds of generalized spectral problems and without any possibility of achieving completeness, we refer, for instance, to [2], [12], [55], [68], [72], [73], [74], [75], [98], [116], [131], and the extensive literature cited therein in the context of general boundary value problems. In the context of indefinite Sturm-Liouville-type boundary value problems we mention, for instance, [6], [8], [11], [13], [14], [15], [16], [18], [19], [20], [21], [23], [30], [31], [32], [37], [47], [52], [56], [83], [84], [86], [89], [90], [91], [99], [100], [117], [132], [135,Chs.…”

Section: Introductionmentioning

confidence: 99%

Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

Gesztesy

Weikard

2014

Operator Theory: Advances and Applications

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Dedicated with great pleasure to Ludwig Streit on the occasion of his 75th birthday.Abstract. We study particular cases of left-definite eigenvalue problems Aψ = λBψ, with A ≥ εI for some ε > 0 and B self-adjoint, but B not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where A represents a positive definite Schrödinger or Sturm-Liouville operator T in L 2 (R; dx) associated with a differential expression of the formrepresents an operator of multiplication by r(x) in L 2 (R; dx), which, in general, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. on R. In fact, our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients q and r and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space L 2 (R; dx). Our approach relies on rewriting the eigenvalue problem Aψ = λBψ in the formand a careful study of (appropriate realizations of) the operator A −1/2 BA −1/2 in L 2 (R; dx).In the course of our treatment, we review and employ various necessary and sufficient conditions for q to be relatively bounded (resp., compact) and relatively form bounded (resp., form compact) with respect to T 0 = −d 2 /dx 2 defined on H 2 (R). In addition, we employ a supersymmetric formalism which permits us to factor the second-order operator T into a product of two firstorder operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients q and r, where q may be a distribution and r generates a measure and hence no smoothness is assumed for q and r.

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From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a twodimensional domain with L 2 cells. For fixed A, and L → ∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution equilibrates along stream lines.In this paper, we show that if both A → ∞ and L → ∞, then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves likeσ(A)/L 2 , whereσ(A) ≈ √ AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

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On the asymptotic behavior of the principal eigenvalues of some elliptic problems

Cited by 9 publications

References 11 publications

A functional approach towards eigenvalue problems associated with incompressible flow

A functional approach towards eigenvalue problems associated with incompressible flow

Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

From homogenization to averaging in cellular flows

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