Estimation on the first eigenvalue for some nonlinear Dirichlet eigenvalue problems

2009 ◽  
Vol 71 (12) ◽  
pp. e2442-e2448 ◽  
Author(s):  
Gabriella Bognár
Keyword(s):  
Eigenvalue Problems ◽  
First Eigenvalue ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

scholarly journals The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan-Hsiou Cheng
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Liouville Property ◽  
The First Eigenvalue ◽  
Eigenvalue Gap ◽  
Single Well ◽  
Eigenvalue Ratio ◽  
Sturm Liouville ◽  
Prüfer Substitution

AbstractIn this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

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.

scholarly journals ON THE UPPER ESTIMATES FOR THE FIRST EIGENVALUE OF A STURM - LIOUVILLE PROBLEM WITH A WEIGHTED INTEGRAL CONDITION

2017 ◽  
Vol 21 (6) ◽  
pp. 124-129
Author(s):  
M.Yu. Telnova
Keyword(s):  
Weight Function ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Integral Condition ◽  
First Eigenvalue ◽  
Second Order Differential Equations ◽  
The First Eigenvalue ◽  
Lagrange Problem ◽  
Sturm Liouville ◽  
The Given

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved. In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.

scholarly journals A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold

1994 ◽  
Vol 56 (2) ◽  
pp. 267-277
Author(s):  
Francisco J. Carreras ◽  
Fernando Giménez ◽  
Vicente Miquel
Keyword(s):  
Outer Radius ◽  
Holomorphic Sectional Curvature ◽  
First Eigenvalue ◽  
Kaehler Manifold ◽  
Dirichlet Eigenvalue ◽  
Totally Geodesic Submanifold ◽  
The First Eigenvalue ◽  
First Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalue Problem ◽  
Sharp Lower Bound

AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

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}.

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