Asymptotic properties of a ϕ -Laplacian and Rayleigh quotient

Waldo Arriagada; Jorge Huentutripay

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 3, page 345-362
  • ISSN: 0010-2628

Abstract

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In this paper we consider the ϕ -Laplacian problem with Dirichlet boundary condition, - div ϕ ( | ∇ u | ) ∇ u | ∇ u | = λ g ( · ) ϕ ( u ) in Ω , λ ∈ ℝ and u | ∂ Ω = 0 . The term ϕ is a real odd and increasing homeomorphism, g is a nonnegative function in L ∞ ( Ω ) and Ω ⊆ ℝ N is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer and an eigenfunction of the ϕ -Laplacian. It turns out that if, in addition, a suitable Δ 2 -condition holds then any number greater than or equal to the minimum of the Rayleigh quotient is an eigenvalue of the differential equation.

How to cite

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Arriagada, Waldo, and Huentutripay, Jorge. "Asymptotic properties of a $\varphi $-Laplacian and Rayleigh quotient." Commentationes Mathematicae Universitatis Carolinae 61.3 (2020): 345-362. <http://eudml.org/doc/297143>.

@article{Arriagada2020,
abstract = {In this paper we consider the $\varphi \,$-Laplacian problem with Dirichlet boundary condition, \[ -\{\rm div\}\Big (\varphi (|\nabla u|) \frac\{\nabla u\}\{|\nabla u |\}\Big )=\lambda g(\cdot ) \varphi (u) \qquad \text\{in \} \Omega , \lambda \in \{\mathbb \{R\}\} \text\{ and \} u\vert \_\{\partial \Omega \}=0. \] The term $\varphi $ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^\{\infty \}(\Omega )$ and $\Omega \subseteq \mathbb \{R\}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer and an eigenfunction of the $\varphi $-Laplacian. It turns out that if, in addition, a suitable $\Delta _2$-condition holds then any number greater than or equal to the minimum of the Rayleigh quotient is an eigenvalue of the differential equation.},
author = {Arriagada, Waldo, Huentutripay, Jorge},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Orlicz–Sobolev space; $\varphi $-Laplacian; eigenvalue; Rayleigh quotient},
language = {eng},
number = {3},
pages = {345-362},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Asymptotic properties of a $\varphi $-Laplacian and Rayleigh quotient},
url = {http://eudml.org/doc/297143},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Arriagada, Waldo
AU - Huentutripay, Jorge
TI - Asymptotic properties of a $\varphi $-Laplacian and Rayleigh quotient
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 3
SP - 345
EP - 362
AB - In this paper we consider the $\varphi \,$-Laplacian problem with Dirichlet boundary condition, \[ -{\rm div}\Big (\varphi (|\nabla u|) \frac{\nabla u}{|\nabla u |}\Big )=\lambda g(\cdot ) \varphi (u) \qquad \text{in } \Omega , \lambda \in {\mathbb {R}} \text{ and } u\vert _{\partial \Omega }=0. \] The term $\varphi $ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty }(\Omega )$ and $\Omega \subseteq \mathbb {R}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer and an eigenfunction of the $\varphi $-Laplacian. It turns out that if, in addition, a suitable $\Delta _2$-condition holds then any number greater than or equal to the minimum of the Rayleigh quotient is an eigenvalue of the differential equation.
LA - eng
KW - Orlicz–Sobolev space; $\varphi $-Laplacian; eigenvalue; Rayleigh quotient
UR - http://eudml.org/doc/297143
ER -

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