eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

riccati techniques and discrete oscillations abstract oscillation and nonoscillation criteria are obtained for the second order difference equation r n x n p n x n 1 0 where x n x n 1 x n is the forward difference operator and r n p n are sequences of real numbers the main techniques involved are riccati equations of the first and second levels

superaccurate finite element eigenvalues via a rayleigh quotient correction abstract the consistent finite element formulation of the vibration problem generates upper bounds on the corresponding exact eigenvalues but requires the solution of the highly expensive general algebraic eigenproblem kx mx with a global matrix m that is of the same sparsity pattern as the global stiffness k the lumped diagonal mass matrix finite element formulation is no longer variationally correct but results in a simplified algebraic eigenproblem of comparable accuracy we may write the mass matrix as a linear matrix function m m 1 m 2 of parameter such that m 1 is the diagonal lumped mass matrix and m 0 is the consistent mass matrix it has been shown that an optimal exists between these two states which results in superaccurate eigenvalues what detracts from the appeal of this approach is that the superior accuracy thus achieved comes at the hefty price of having to solve the still general algebraic eigenproblem with a nondiagonal mass matrix in this note we show that the same superior accuracy can be had by first computing an eigenvector u from ku du in which d m 1 m 2 is the lumped diagonal mass matrix and then obtaining the corresponding superaccurate eigenvalue from the rayleigh quotient r u u t ku u t m u m m 1 m 2 for an optimal

blow up analysis of a finsler liouville equation in two dimensions abstract in this paper we establish a blow up analysis for the finsler liouville equation q u v x e u in two dimensions here the operator q is the so called finsler laplacian which is a generalized laplacian as a byproduct we obtain an existence result

q curvature flow on the standard sphere of even dimension using a gradient flow approach initiated by s brendle we generalize the existence theorem for the prescribing q curvature equation on s2 gauss curvature by m struwe 2005 14 and on s4 by malchiodi and struwe 2006 12 to sn for all even n with the similar assumption on the prescribed curvature candidate f

positive solutions of higher order nonlinear delay difference equations abstract classification schemes for positive solutions of a class of higher order nonlinear delay difference equations are given in terms of their asymptotic behavior and necessary as well as sufficient conditions for the existence of these solutions are also provided

nonoscillation of second order linear difference systems with varying coefficients abstract this paper deals with nonoscillation problem about the non autonomous linear difference system x n a n x n 1 n 1 2 where a n is a 2 2 variable matrix that is nonsingular for n n in the special case that a is a constant matrix it is well known that all non trivial solutions are nonoscillatory if and only if all eigenvalues of a are positive real numbers namely det a 0 tr a 0 and det a tr a 2 1 4 the well known result can be said to be an analogy of ordinary differential equations the results obtained in this paper extend this analogy result in other words this paper clarifies the distinction between difference equations and ordinary differential equations our results are explained with some specific examples in addition figures are attached to facilitate understanding of those examples

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

nonoscillation theorems for second order linear difference equations via the riccati type transformation ii the present paper deals with nonoscillation problem for the second order linear difference equation cnxn 1 cn1xn1 bnxn n 1 2 where bn and cn are positive sequences all nontrivial solutions of this equation are nonoscillatory if and only if the riccati type difference equation qnzn 1zn1 1has an eventually positive solution where qn cn2 bnbn 1 our nonoscillation theorems are proved by using this equivalence relation in particular it is focusing on the relation of the triple q3k2 q3k1 q3k for each kn our results can also be applied to not only the case that bn and cn are periodic but also the case that bn or cn is non periodic to compare the obtained results with previous works we give some concrete examples and those simulations

asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations mingshu peng department of applied mathematics beijing institute of technology beijing 100081 p r china qianli xu department of mathematics yiyang teachers college yiyang 413049 p r china lihong huang department of applied mathematics hunan university changsha 410082 p r china weigao ge

a note on the problem of prescribing gaussian curvature on surfaces the problem of existence of conformal metrics with gaussian curvature equal to a given function k on a compact riemannian 2 manifold m of negative euler characteristic is studied let ko be any nonconstant function on m with max ko 0 and let ka ko a it is proved that there exists a a 0 such that the problem has a solution for k ka iff a e oo a moreover if a e 0 a then the problem has at least 2 solutions let m be a closed 2 dimensional smooth manifold and g be a riemannian metric on m let k denote the gaussian curvature of g if g e2ug is another riemannian metric conformal to g and has gaussian curvature k then it is well known that k e 2u k au where a is the laplacian of g given a function k e c m the problem of prescribing gaussian curvature asks whether one can find u e c m such that the metric g e2ug has the given k as its gaussian curvature obviously this is equivalent to the problem of solvability of the following elliptic equation 1 au k ke2u o onm if u is a solution of 1 then we have by integrating 1

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

asymptotic behavior of solutions of second order nonlinear difference equations 0 2 j 1 b n u n g n y y h 1 1 or equivalently e y a0 by g n y ey where e is the operator defined by ey n y n 1 2 let z z2 be a fundamental system of solutions of the linear difference equation lz e2z f aez bz 0 3 satisfying z o 1 z l 0 zz o 0 22 l 1 4 we will prove that there exists a ball b 0 p c r2 such that if y o y 1 e b 0 p then the solution y of equation 1 is defined on all n and for n co it satisfies

a compactness result for periodic multivortices in the electroweak theory we derive a priori uniform bounds for solutions of an elliptic system of liouville type equations first analyzed by j spruck and y yang comm math phys 144 1992 1 yielding periodic multivortices in the classical electroweak theory of glashow salam weinberg our proof is based on a concentration quantization result in the same spirit of brezis merle comm partial differential equations 16 8 9 1991 1223 and li shafrir indiana univ math j 43 4 1994 1255 for mean field equations on riemannian compact 2 manifolds

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

disconjugacy for nth order linear difference equations necessary and sufficient conditions for disconjugacy are given in terms of the coefficients ai 0 i n of p by hartman 3 theorem 5 21 we will give necessary and sufficient conditions in terms of the coefficients ai 0 i g n for right 5 n l disconjugacy and left l n i disconjugacy as defined by peterson 9 a corollary of these results improves those in hartman s paper in one direction a special case of these results will generalize the results in 4 a corollary to these theorems will be the equivalence of right i n i disconjugacy and left 5 n i disconjugacy provided condition 2 holds which generalizes results of hankerson 2

nonoscillation oscillation and growth of solutions of nonlinear difference equations of second order abstract sufficient conditions are obtained which guarantee that all solutions or all bounded solutions of a class of second order nonlinear difference equations are nonoscillatory oscillation criteria and theorems concerning the growth of solutions are also given

oscillation criteria for two dimensional difference systems of first order linear difference equations sufficient conditions are established for the oscillation of the linear two dimensional difference system dx n p ny n dy n 1 q nx n n n n 0 n 0 n 0 1 where p n q n are nonnegative real sequences our results extend the results in the literature

conformal metrics with prescribed gaussian curvature on s2 we consider on s 2 the problem of which functions k can be the scalar curvature of a metric conformal to the standard metric on s 2 we assume that k is a function of one variable and we obtain a necessary and sufficient condition for the problem to be solvable we also obtain several new sufficient conditions on k which are easy to check in order that the problem be solvable

oscillation results for second order scalar and matrix difference equations we consider the self adjoint second order scalar difference equation rn xn pnxn 1 0 1 and the matrix system rn xn pnxn 1 0 2 where rn 0 pn 0 rn 0 pn 0 are sequences of real numbers d d hermitian matrices with rn 0 rn 0 we obtain some new oscillation criteria through change of variable techniques combined with consideration of 1 and 2 as perturbations of the cases pn 0 pn 0

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

smoothness of solutions for delay difference equations abstract for the delay difference equation x n 1 f n x n x n n satisfying x j j j belonging to an initial segment smooth dependence of a solution x n with respect to is considered

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

oscillation theorems for second order neutral difference equations abstract oscillation criteria for the second order difference equations of the form 2xn qnf xn r 0 and the neutral difference equation 2 xn pxn k qnf xn r 0 are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn is a sequence of nonnegtive real numbers and f r r is continuous

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

a classification of the solutions of a difference equation according to their behavior at infinity on considere l equation aux differences lineaires d ordre n ly t p t y t 0 ou p t est de signe defini sur a a a 1 a est un entier

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

unbounded positive solutions of higher order difference equations with singular nonlinear term in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form mxn 1 f n xn 0 nen0 where m 2 1 or 1 and f n0 r r is a real valued continuous function such that f n x 0 for all n x n0 0 where n0 0 1

exponential elliptic boundary value problems on a solid torus in the critical of supercritical case abstract in this paper we investigate the behavior and the existence of positive and non radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus t of r 3 when data are invariant under the group g o 2 i o 3 the model problems of interest are stated below f x e 0 on t t 0 and a f e 0 0 on t n b g e 0 on t we prove that exist solutions which are g invariant and these exhibit no radial symmetries in order to solve the above problems we need to find the best constants in the sobolev inequalities in the exceptional case

oscillation of linear difference equations with deviating arguments the following difference equation with deviating arguments is considered where u k u k 1 u k 2 pj j 1 m is a sequence of nonnegative numbers j and limk j k j 1 m in the paper sufficient conditions are established for all proper solutions of the above equation to be oscillatory

a small note on symmetric geodesic curvature on d2 some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

bounded oscillation criteria for second order nonlinear delay difference equations of unstable type this paper deals with some sufficient conditions for the bounded oscillation criteria for the second order nonlinear delay difference equation d 2x email protected i 1mp ix n k i f n x n l 1 x n l s n 0 1 2 in addition we generalize and improve the existing results and then give some examples of applications

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

a discrete interpretation of reid s roundabout theorem for generalized differential systems abstract in a series of papers starting with a 1959 paper in j math mechanics 1 w t reid presented sturmian theory and asymptotic behavior for generalized differential systems these systems were equivalent to a type of linear vector riemann stieltjes integral equation reid s primary result was his roundabout theorem for this generalized setting as he pointed out if the measure is piecewise constant then results for difference equations ensue the objectives of this study are 1 i to interpret reid s results for both jacobi and riccati difference equations and 2 ii to compare those results with subsequent studies of difference equations based on discrete variational theory

bounded and zero convergent solutions of second order difference equations abstract in this paper we study boundedness and monotonicity properties of a homogeneous second order linear difference equation under certain conditions it is shown that every solution must eventually be monotonic the question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated a necessary and sufficient condition for all solutions to be bounded is presented comparison theorems are also discussed

oscillation theory and renormalized oscillation theory for jacobi operators we provide a comprehensive treatment of oscillation theory for jacobi operators with separated boundary conditions our main results are as follows ifusolves the jacobi equation hu n a n u n 1 a n 1 u n 1 b n u n u n r in the weak sense on an arbitrary interval and satisfies the boundary condition on the left or right then the dimension of the spectral projectionp h ofhequals the number of nodes i e sign flips ifa n 0 ofu moreover we present a reformulation of oscillation theory in terms of wronskians of solutions thereby extending the range of applicability for this theory if 1 2 r and ifu1 2solve the jacobi equationhuj juj j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projectionp 1 2 h equals the number of nodes of the wronskian ofu1andu2 furthermore these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic jacobi operators