Universal inequalities for eigenvalues of a system of elliptic equations

2009 ◽  
Vol 139 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Hong-Cang Yang
Keyword(s):  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Lower Order ◽  
Elliptic Equations ◽  
Elastic Vibration ◽  
Dimensional Euclidean Space ◽  
Explicit Method ◽  
Universal Inequalities ◽  
Image Position

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

Oscillation of Elliptic Equations in General Domains

1975 ◽  
Vol 27 (6) ◽  
pp. 1239-1245 ◽  
Author(s):  
E. S. Noussair
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Euclidean Space ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
General Type ◽  
Elliptic Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Image Position

Oscillation criteria will be obtained for the linear elliptic partial differential equationin an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by where each α (i), i = 1, … , n, is a non-negative integer.

INEQUALITIES FOR EIGENVALUES OF LAPLACIAN WITH ANY ORDER

2009 ◽  
Vol 11 (04) ◽  
pp. 639-655 ◽  
Author(s):  
QING-MING CHENG ◽  
TAKAMICHI ICHIKAWA ◽  
SHINJI MAMETSUKA
Keyword(s):  
Euclidean Space ◽  
Bounded Domain ◽  
Lower Order ◽  
Dimensional Euclidean Space

In this paper, we study eigenvalues of Laplacian with any order on a bounded domain in an n-dimensional Euclidean space and obtain estimates for eigenvalues, which are the Yang-type inequalities. In particular, the sharper result of Yang is included here. Furthermore, for lower order eigenvalues, we obtain two sharper inequalities. As a consequence, a proof of results announced by Ashbaugh [1] is also given.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

A Note on a Comparison Result for Elliptic Equations

1977 ◽  
Vol 29 (5) ◽  
pp. 1081-1085 ◽  
Author(s):  
W. Allegretto
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Comparison Theorem ◽  
Elliptic Equations ◽  
Comparison Result ◽  
Mixed Derivatives ◽  
Image Position ◽  
Derivatives Of

In a recent paper [2], Bushard established and applied a comparison theorem for positive solutions to the equation:in an arbitrary bounded domain D of Euclidean w-space Rn. The proof of these results depended on the absence of mixed derivatives of u in the equation considered.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

Symmetric Forms

1970 ◽  
Vol 13 (1) ◽  
pp. 83-87 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Image Position ◽  
Class Of Functions

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by1.1

The number of polygons on a lattice

1961 ◽  
Vol 57 (3) ◽  
pp. 516-523 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
End Points ◽  
Unit Distance ◽  
Connective Constant ◽  
Image Position ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such thatHere I shall prove the truth of the long-standing conjecture thatI shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.

scholarly journals Lower bounds for fractional Laplacian eigenvalues

2014 ◽  
Vol 16 (06) ◽  
pp. 1450032
Author(s):  
Guoxin Wei ◽  
He-Jun Sun ◽  
Lingzhong Zeng
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Bounded Domain ◽  
Lower Bounds ◽  
Fractional Laplacian ◽  
Dimensional Euclidean Space ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of The Laplacian ◽  
Eigenvalue Inequality

In this paper, we investigate eigenvalues of fractional Laplacian (–Δ)α/2|D, where α ∈ (0, 2], on a bounded domain in an n-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which improves some results due to Yildirim Yolcu and Yolcu in [Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math. 15(3) (2013) 1250048]. In particular, for the case of Laplacian, we obtain a sharper eigenvalue inequality, which gives an improvement of the result due to Melas in [A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003) 631–636].

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

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