EIGENVALUE ESTIMATES ON DOMAIN OF THE POLYDISK

2012 ◽  
Vol 23 (01) ◽  
pp. 1250014
Author(s):  
TAO ZHENG ◽  
DAGUANG CHEN ◽  
MIN CAI
Keyword(s):  
Bounded Domain ◽  
First Eigenvalue ◽  
Dirichlet Laplacian ◽  
Clamped Plate ◽  
Eigenvalue Estimates ◽  
The First Eigenvalue ◽  
Universal Inequalities

In this paper, we investigate universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem on a bounded domain in an n-dimensional polydisk 𝔻n. Moreover, from the domain monotonicity of the eigenvalue, we can prove that if the first eigenvalue of the Dirichlet Laplacian tends to [Formula: see text] when the domain tends to the polydisk 𝔻n, then all of the eigenvalues tend to [Formula: see text].

scholarly journals Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marcello Lucia ◽  
Guido Sweers
Keyword(s):  
Bounded Domain ◽  
Cooperative Systems ◽  
First Eigenvalue ◽  
Exponential Nonlinearity ◽  
The First Eigenvalue ◽  
Chern Simons ◽  
Fully Coupled

<p style='text-indent:20px;'>We consider fully coupled cooperative systems on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with exponential nonlinearity, are nondegenerate.</p>

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.

An eigenvalue optimization problem for the p-Laplacian

2015 ◽  
Vol 145 (6) ◽  
pp. 1145-1151 ◽  
Author(s):  
Anisa M. H. Chorwadwala ◽  
Rajesh Mahadevan
Keyword(s):  
Eigenvalue Problem ◽  
Optimization Problem ◽  
First Eigenvalue ◽  
Eigenvalue Optimization ◽  
Nonlinear Case ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Monotonicity Result ◽  
Open Question

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

scholarly journals On relations between principal eigenvalue and torsional rigidity

2020 ◽  
pp. 2050093
Author(s):  
Michiel van den Berg ◽  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
Optimization Problem ◽  
Torsional Rigidity ◽  
Principal Eigenvalue ◽  
Higher Dimensions ◽  
First Eigenvalue ◽  
Thin Domains ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Dimension One

We consider the problem of minimizing or maximizing the quantity [Formula: see text] on the class of open sets of prescribed Lebesgue measure. Here [Formula: see text] is fixed, [Formula: see text] denotes the first eigenvalue of the Dirichlet Laplacian on [Formula: see text], while [Formula: see text] is the torsional rigidity of [Formula: see text]. The optimization problem above is considered in the class of all domains [Formula: see text], in the class of convex domains [Formula: see text], and in the class of thin domains. The full Blaschke–Santaló diagram for [Formula: see text] and [Formula: see text] is obtained in dimension one, while for higher dimensions we provide some bounds.

scholarly journals A spectral shape optimization problem with a nonlocal competing term

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Dario Mazzoleni ◽  
Berardo Ruffini
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Critical Value ◽  
Relative Strength ◽  
Spectral Shape ◽  
First Eigenvalue ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Existence Of Minimizers ◽  
Type Interaction

AbstractWe study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

scholarly journals The existence of nontrivial solutions for a critically coupled Schr\”{o}dinger system in a bounded domain of $\R^3$

2021 ◽  
Author(s):  
Hongyu Ye ◽  
Lina Zhang
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
The First Eigenvalue ◽  
Nirenberg Problem

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

scholarly journals A Class of Equations with Three Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 478 ◽  
Author(s):  
Biagio Ricceri
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Convex Set ◽  
First Eigenvalue ◽  
Global Minima ◽  
Three Solutions ◽  
Smooth Bounded Domain ◽  
The First Eigenvalue

Here is one of the results obtained in this paper: Let Ω ⊂ R n be a smooth bounded domain, let q > 1 , with q < n + 2 n - 2 if n ≥ 3 and let λ 1 be the first eigenvalue of the problem - Δ u = λ u in Ω , u = 0 on ∂ Ω . Then, for every λ > λ 1 and for every convex set S ⊆ H 0 1 ( Ω ) dense in H 0 1 ( Ω ) , there exists α ∈ S such that the problem - Δ u = λ ( u + - ( u + ) q ) + α ( x ) in Ω , u = 0 on ∂ Ω , has at least three weak solutions, two of which are global minima in H 0 1 ( Ω ) of the functional u → 1 2 ∫ Ω | ∇ u ( x ) | 2 d x - λ ∫ Ω 1 2 | u + ( x ) | 2 - 1 q + 1 | u + ( x ) | q + 1 d x - ∫ Ω α ( x ) u ( x ) d x where u + = max { u , 0 } .

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