Lower Bounds for the First Eigenvalue of Elliptic Equations

1960 ◽  
Vol 71 (3) ◽  
pp. 423 ◽  
Author(s):  
M. H. Protter
Keyword(s):  
Lower Bounds ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
The First Eigenvalue

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

scholarly journals Principal frequency of the p-Laplacian and the inradius of Euclidean domains

2015 ◽  
Vol 07 (03) ◽  
pp. 505-511 ◽  
Author(s):  
Guillaume Poliquin
Keyword(s):  
Lower Bound ◽  
Laplace Operator ◽  
Lower Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Principal Frequency ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Euclidean Domains

We study the lower bounds for the principal frequency of the p-Laplacian on N-dimensional Euclidean domains. For p > N, we obtain a lower bound for the first eigenvalue of the p-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if p > N - 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.

A MULTIPLICITY RESULT FOR ELLIPTIC EQUATIONS AT CRITICAL GROWTH IN LOW DIMENSION

2003 ◽  
Vol 05 (02) ◽  
pp. 171-177 ◽  
Author(s):  
GIUSEPPE DEVILLANOVA ◽  
SERGIO SOLIMINI
Keyword(s):  
Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Multiplicity Result ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Low Dimension ◽  
Sobolev Exponent ◽  
Regular Subset

We consider the problem -Δu = |u|2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝN (N ≥ 3), [Formula: see text] is the critical Sobolev exponent and λ is a constant in ]0, λ1[ where λ1 is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least [Formula: see text] (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

On a maximum principle for a class of fourth-order semilinear elliptic equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 319-323 ◽  
Author(s):  
Philip W. Schaefer
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Linear Partial Differential Equation ◽  
First Eigenvalue ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Maximum Value ◽  
Non Linear ◽  
Dirichlet Eigenvalue Problem

SynopsisIt is shown that Ф = | grad u |2–uΔu, where u is a solution of Δ2u+pf(u) = 0 in D, assumes its maximum value on the boundary of D. This principle leads one to a lower bound on the first eigenvalue in the non-linear Dirichlet eigenvalue problem and to the non-existence of solutions to this non-linear partial differential equation subject to certain zero boundaryconditions.

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