Boundary eigencurve problems involving the biharmonic operator

Omar Chakrone; Najib Tsouli; Mostafa Rahmani; Omar Darhouche

Applicationes Mathematicae (2014)

  • Volume: 41, Issue: 2-3, page 267-275
  • ISSN: 1233-7234

Abstract

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The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem ⎧Δ²u = f(u,x) + βΔu + h in Ω, ⎨ ⎩Δu = u = 0 ∂Ω, where f: Ω × ℝ → ℝ is a Carathéodory function and h is a given function in L²(Ω).

How to cite

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Omar Chakrone, et al. "Boundary eigencurve problems involving the biharmonic operator." Applicationes Mathematicae 41.2-3 (2014): 267-275. <http://eudml.org/doc/280083>.

@article{OmarChakrone2014,
abstract = { The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem ⎧Δ²u = f(u,x) + βΔu + h in Ω, ⎨ ⎩Δu = u = 0 ∂Ω, where f: Ω × ℝ → ℝ is a Carathéodory function and h is a given function in L²(Ω). },
author = {Omar Chakrone, Najib Tsouli, Mostafa Rahmani, Omar Darhouche},
journal = {Applicationes Mathematicae},
keywords = {biharmonic operator; eigencurve; resonance},
language = {eng},
number = {2-3},
pages = {267-275},
title = {Boundary eigencurve problems involving the biharmonic operator},
url = {http://eudml.org/doc/280083},
volume = {41},
year = {2014},
}

TY - JOUR
AU - Omar Chakrone
AU - Najib Tsouli
AU - Mostafa Rahmani
AU - Omar Darhouche
TI - Boundary eigencurve problems involving the biharmonic operator
JO - Applicationes Mathematicae
PY - 2014
VL - 41
IS - 2-3
SP - 267
EP - 275
AB - The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem ⎧Δ²u = f(u,x) + βΔu + h in Ω, ⎨ ⎩Δu = u = 0 ∂Ω, where f: Ω × ℝ → ℝ is a Carathéodory function and h is a given function in L²(Ω).
LA - eng
KW - biharmonic operator; eigencurve; resonance
UR - http://eudml.org/doc/280083
ER -

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