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Displaying similar documents to “Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole”

Asymptotic properties of a ϕ -Laplacian and Rayleigh quotient

Waldo Arriagada, Jorge Huentutripay (2020)

Commentationes Mathematicae Universitatis Carolinae

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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 and u | Ω = 0 . The term ϕ is a real odd and increasing homeomorphism, g is a nonnegative function in L ( Ω ) and Ω 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...

Asymptotic integration of differential equations with singular p -Laplacian

Milan Medveď, Eva Pekárková (2016)

Archivum Mathematicum

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In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with p - Laplacian, where 1 < p < 2 . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as t .

Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

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We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p -Laplacian and the Navier p -biharmonic operator on a ball of radius R in N and its asymptotics for p approaching 1 and . Let p tend to . There is a critical radius R C of the ball such that the principal eigenvalue goes to for 0 < R R C and to 0 for R > R C . The critical radius is R C = 1 for any N for the p -Laplacian and R C = 2 N in the case of the p -biharmonic operator. When p approaches 1 , the principal eigenvalue...

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of a positive solution of the system ⎧ - p ( x ) u = F ( x , u , v ) , x ∈ Ω, ⎨ - q ( x ) v = H ( x , u , v ) , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where Ω N is a bounded domain with C² boundary, F ( x , u , v ) = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] , H ( x , u , v ) = λ q ( x ) [ g ( x ) b ( v ) + h ( u ) ] , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and - p ( x ) u = - d i v ( | u | p ( x ) - 2 u ) is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Michael Gil&#039; (2024)

Czechoslovak Mathematical Journal

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Let A be a bounded linear operator in a complex separable Hilbert space , and S be a selfadjoint operator in . Assuming that A - S belongs to the Schatten-von Neumann ideal 𝒮 p ( p > 1 ) , we derive a bound for k | R λ k ( A ) - λ k ( S ) | p , where λ k ( A ) ( k = 1 , 2 , ) are the eigenvalues of A . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case p = 2 we refine the Weyl inequality between the real parts of the eigenvalues of A and the eigenvalues...

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang (2021)

Czechoslovak Mathematical Journal

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We construct some nondecreasing quantities associated to the first eigenvalue of - Δ φ + c R ( c 1 2 ( n - 2 ) / ( n - 1 ) ) along the Yamabe flow, where Δ φ is the Witten-Laplacian operator with a C 2 function φ . We also prove a monotonic result on the first eigenvalue of - Δ φ + 1 4 ( n / ( n - 1 ) ) R along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of - Δ φ + c R a with a ( 0 , 1 ) along the Yamabe flow.

Global continuum of positive solutions for discrete p -Laplacian eigenvalue problems

Dingyong Bai, Yuming Chen (2015)

Applications of Mathematics

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We discuss the discrete p -Laplacian eigenvalue problem, Δ ( φ p ( Δ u ( k - 1 ) ) ) + λ a ( k ) g ( u ( k ) ) = 0 , k { 1 , 2 , ... , T } , u ( 0 ) = u ( T + 1 ) = 0 , where T > 1 is a given positive integer and φ p ( x ) : = | x | p - 2 x , p > 1 . First, the existence of an unbounded continuum 𝒞 of positive solutions emanating from ( λ , u ) = ( 0 , 0 ) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum 𝒞 is a monotone continuous curve globally defined for all λ > 0 .

Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure η supported on a compact subset of U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator η and a second order differential operator Δ η , with respect to η . We generalize this approach to measures of the form η : = ν + δ , where ν is non-atomic and δ is finitely supported. We determine...

Inequalities for real number sequences with applications in spectral graph theory

Emina Milovanović, Şerife Burcu Bozkurt Altındağ, Marjan Matejić, Igor Milovanović (2022)

Czechoslovak Mathematical Journal

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Let a = ( a 1 , a 2 , ... , a n ) be a nonincreasing sequence of positive real numbers. Denote by S = { 1 , 2 , ... , n } the index set and by J k = { I = { r 1 , r 2 , ... , r k } , 1 r 1 < r 2 < < r k n } the set of all subsets of S of cardinality k , 1 k n - 1 . In addition, denote by a I = a r 1 + a r 2 + + a r k , 1 k n - 1 , 1 r 1 < r 2 < < r k n , the sum of k arbitrary elements of sequence a , where a I 1 = a 1 + a 2 + + a k and a I n = a n - k + 1 + a n - k + 2 + + a n . We consider bounds of the quantities R S k ( a ) = a I 1 / a I n , L S k ( a ) = a I 1 - a I n and S k , α ( a ) = I J k a I α in terms of A = i = 1 n a i and B = i = 1 n a i 2 . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let { X i j } , i , j = , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

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We consider the semilinear Lane–Emden problem where p > 1 and Ω is a smooth bounded domain of 2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( p ) , as p + . Among other results we show, under some symmetry assumptions on Ω , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p + , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...