Optimization of the First Eigenvalue of Equations with Indefinite Weights

2013 ◽  
Vol 13 (1) ◽  
Author(s):  
Chris Cosner ◽  
Fabrizio Cuccu ◽  
Giovanni Porru
Keyword(s):  
Dirichlet Boundary ◽  
Optimization Problems ◽  
Principal Eigenvalue ◽  
Practical Importance ◽  
Spatial Arrangement ◽  
Reserve Design ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Indefinite Weights

AbstractWe investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.

scholarly journals Hemivariational inequalities with the potential crossing the first eigenvalue

2001 ◽  
Vol 64 (3) ◽  
pp. 381-393 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Dirichlet Boundary ◽  
Mountain Pass Theorem ◽  
Principal Eigenvalue ◽  
Dirichlet Boundary Conditions ◽  
Hemivariational Inequalities ◽  
Linking Theorem ◽  
First Eigenvalue ◽  
The First Eigenvalue

In this paper we study a nonlinear hemivariational inequality involving the p-Laplacian. Our approach is variational and uses a recent nonsmooth Linking Theorem, due to Kourogenis and Papageorgiou (2000). The use of the Linking Theorem instead of the Mountain Pass Theorem allows us to assume an asymptotic behaviour of the generalised potential function which goes beyond the principal eigenvalue of the negative p-Laplacian with Dirichlet boundary conditions.

LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5

2009 ◽  
Vol 11 (01) ◽  
pp. 59-69 ◽  
Author(s):  
PAOLO ROSELLI ◽  
MICHEL WILLEM
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Nodal Solutions ◽  
The First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Nirenberg Problem ◽  
Least Energy

We prove the existence of (a pair of) least energy sign changing solutions of [Formula: see text] when Ω is a bounded domain in ℝN, N = 5 and λ is slightly smaller than λ1, the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions on Ω.

scholarly journals Comparison estimates on the first eigenvalue of a quasilinear elliptic system

2020 ◽  
Vol 26 (2) ◽  
pp. 273-285
Author(s):  
Abimbola Abolarinwa ◽  
Shahroud Azami
Keyword(s):  
Riemannian Manifolds ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Quasilinear Elliptic System ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Cheeger’S Constant ◽  
Quasilinear Elliptic

AbstractWe study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber–Krahn for the first eigenvalue of a {(p,q)}-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, {1<p<\infty}, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a {(p,q)}-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as {p,q\to 1,1}.

scholarly journals A variation on the Donsker–Varadhan inequality for the principal eigenvalue

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160877
Author(s):  
Jianfeng Lu ◽  
Stefan Steinerberger
Keyword(s):  
Lower Bound ◽  
Exit Time ◽  
Principal Eigenvalue ◽  
Short Paper ◽  
First Eigenvalue ◽  
Drift Diffusion ◽  
First Exit Time ◽  
The First Eigenvalue ◽  
Order Elliptic Operator ◽  
The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

scholarly journals Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

scholarly journals Extinction and Nonextinction for the Fast Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chunlai Mu ◽  
Li Yan ◽  
Yi-bin Xiao
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Dirichlet Boundary ◽  
Fast Diffusion ◽  
First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Fast Diffusion Equation ◽  
The First Eigenvalue

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain ofRNwithN>2. For0<m<1, under appropriate hypotheses, we show thatm=pis the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of-Δwith homogeneous Dirichlet boundary.

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