a second eigenvalue bound for the dirichlet schrodinger operator let i v be the ith eigenvalue of the schrodinger operator with dirichlet boundary conditions on a bounded domain omega subset mathbb r n and with the positive potential v following the spirit of the payne polya weinberger conjecture and under some convexity assumptions on the spherically rearranged potential v we prove that 2 v 2 s1 v here s1 denotes the ball centered at the origin that satisfies the condition 1 v 1 s1 v

an optimal order collocation method for first kind boundary integral equations on polygons this paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in bbb r 2 before discretization the equation is transformed to an equivalent equation over pi pi using a nonlinear parametrization of the polygon which varies more slowly than arc length near each corner this has the effect of producing a transformed equation with a solution which is smooth on pi pi this latter integral equation is shown to be well posed in appropriate sobolev spaces the structure of the integral operator is described in detail and can be written in terms of certain non standard mellin convolution operators using this information we are able to show that the collocation method using splines of order k degree k 1 converges with optimal order o h k the collocation points are the mid points of subintervals when k is odd and the break points when k is even and stability is shown under the assumption that the method may be modified slightly using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method the convergence results are illustrated with some numerical examples

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

a numerical study of the spectral gap we present a numerical study of the spectral gap of the dirichlet laplacian k 2 k 1 k of a planar convex region k besides providing supporting numerical evidence for the long standing gap conjecture that k 3 2 d2 k where d k denotes the diameter of k our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter but also on the perimeter and the area one of these conjectures is a stronger version of the gap conjecture mentioned above a similar study is carried out for the quotient of the first two dirichlet eigenvalues

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

an isoperimetric problem for point interactions we consider a hamiltonian with n point interactions in all with the same coupling constant placed at vertices of an equilateral polygon it is shown that the ground state energy is locally maximized by a regular polygon the question whether the maximum is global is reduced to an interesting geometric problem

inequalities for eigenvalues of fourth order elliptic operators in divergence form on riemannian manifolds abstract in this paper we study eigenvalue of linear fourth order elliptic operators in divergence form with dirichlet boundary condition on a bounded domain in a compact riemannian manifolds with boundary possibly empty and provide a general inequality for them as an application by using this inequality we study eigenvalues of this operator on compact domains of complete submanifolds in a euclidean space

numerical methods for integral equations of mellin type we present a survey of numerical methods based on piecewise polynomial approximation for integral equations of mellin type including examples arising in boundary integral methods for partial differential equations on polygonal domains

mesh grading for integral equations of the first kind with logarithmic kernel when the galerkin method is used to solve the first kind integral equations with logarithmic kernel on polygons or open arcs the singularities of the solution degrade the rates of convergence the aim of this paper is to show for the particular case of piecewise constant approximation spaces that in some cases the o h 3 order of convergence can be restored by a suitable grading of the mesh

isoperimetric bounds for higher eigenvalue ratios for the n dimensional fixed membrane problem we give several results which extend our recent proof of the payne polya weinberger conjecture to ratios of higher eigenvalues in particular we show that for a bounded domain n the eigenvalues of its dirichlet laplacian obey where m denotes the m th eigenvalue and j p k denotes the k th positive zero of the bessel function j p x certain extensions of this result are given the most general being the bound where k 2 and l m denotes the number of nodal domains of an m th eigenfunction our results imply certain further conjectures of payne polya and weinberger concerning 3 2 and 4 3 in addition we find a resonably good bound on 4 1 we also briefly discuss extensions to schrodinger operators and other elliptic eigenvalue problems

on trace identities and universal eigenvalue estimates for some partial differential operators in this article we prove and exploit a trace identity for the spectra of schrodinger operators and similar operators this identity leads to universal bounds on the spectra which apply to low lying eigenvalues eigenvalue asymptotics and to partition functions traces of heat operators in many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated special cases corresponding to known inequalities include those of hile and protter and of yang introduction in this article we prove and exploit an identity for the spectra of self adjoint operators h modeled on the dirichlet laplacian or more generally on schrodinger operators of the form p a x 2 v x 1 where p 1i is the usual momentum operator in convenient units and a x is the magnetic vector potential we recover and extend several known inequalities involving sums differences and ratios of eigenvalues let j j 1 denote the ordered eigenvalues of the dirichlet laplacian on a bounded d dimensional domain with zero dirichlet boundary conditions and recall that hile and protter hipr80 proved that d 4 1 n n j 1 j n 1 j 2 thereby extending an earlier inequality of payne polya and weinberger papowe56 in the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of 2 a cartesian coordinate xj and the corresponding derivative xj for this analysis and various extensions of 2 see ha88 ho90 ha93 hami95 while inequalities of this type have been fairly sharp for low lying eigenvalues they have been mostly disappointing for higher eigenvalues yang ya91 however received by the editors september 28 1995 1991 mathematics subject classification primary 35j10 35j25 58g25

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes we consider the following eigenvalue optimization problem given a bounded domain and numbers 0 a 0 find a subset d of area a for which the first dirichlet eigenvalue of the operator d is as small as possible

an isoperimetric problem for leaky loops and related mean chord inequalities we consider a class of hamiltonians in l2 r2 with attractive interaction supported by piecewise c2 smooth loops of a fixed length l formally given by x with 0 it is shown that the ground state of this operator is locally maximized by a circular we also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of

trace identities for commutators with applications to the distribution of eigenvalues we prove trace identities for commutators of operators which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic schrodinger operators and schrodinger operators on immersed manifolds in particular we prove bounds on the eigenvalue lambda n 1 in terms of the lower spectrum bounds on ratios of means of eigenvalues and universal monotonicity properties of eigenvalue moments which imply sharp versions of lieb thirring inequalities in the case of a schrodinger operator on an immersed manifold of dimension d we derive a version of reilly s inequality bounding the eigenvalue lambda n 1 of the laplace beltrami operator by a universal constant times vertical bar vertical bar h vertical bar n 2 infinity 2 d

global polynomial approximation for symm s equation on polygons summary to solve 1d linear integral equations on bounded intervals with nonsmooth input functions and solutions we have recently proposed a quite general procedure that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials in particular the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap in these cases we have obtained arbitrarily high orders of convergence through the solution of very well conditioned linear systems in this paper to enlarge the domain of applicability of our technique we show how the above procedure can be successfully used also to solve the classical symm s equation on a piecewise smooth curve the collocation method we propose applied to the transformed equation and based on chebyshev polynomials of the first kind has shown to be stable and convergent a comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive

eigenvalue inequalities for klein gordon operators we consider the pseudodifferential operators hm associated by the prescriptions of quantum mechanics to the klein gordon hamiltonian p 2 m2 when restricted to a bounded open domain rd when the mass m is 0 the operator h0 coincides with the generator of the cauchy stochastic process with a killing condition on the operator h0 is sometimes called the fractional laplacian with power 12 cf r banuelos t kulczycki eigenvalue gaps for the cauchy process and a poincare inequality j funct anal 211 2 2004 355 423 r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 234 2006 199 225 e giere the fractional laplacian in applications http www eckhard giere de math publications review pdf we prove several universal inequalities for the eigenvalues 0 1 2 of hm and their means k 1k l 1k l among the inequalities proved are k cst k 1 d for an explicit optimal semiclassical constant depending only on the dimension d for any dimension d 2 and any k k 1 d 1d 1 k furthermore when d 2 and k 2j k j d21 d d 1 kj 1d finally we present some analogous estimates allowing for an operator including an external potential energy field i e hm v x for v x in certain function classes

estimates for eigenvalues on riemannian manifolds abstract in this paper we investigate eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in an n dimensional complete riemannian manifold m when m is an n dimensional euclidean space r n the conjecture of polya is well known the kth eigenvalue k of the dirichlet eigenvalue problem of laplacian satisfies k 4 2 n vol 2 n k 2 n for k 1 2 li and yau p li s t yau on the schrodinger equation and the eigenvalue problem comm math phys 88 1983 309 318 cf lieb e lieb the number of bound states of one body schrodinger operators and the weyl problem in proc sympos pure math vol 36 1980 pp 241 252 have given a partial solution for the conjecture of polya that is they have proved 1 k i 1 k i n n 2 4 2 n vol 2 n k 2 n for k 1 2 which is sharp in the sense of average in this paper we consider a general setting for complete riemannian manifolds we establish an analog of the li and yau s inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a complete riemannian manifold furthermore we obtain a universal inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a hyperbolic space h n 1 from it we prove that when the bounded domain tends to h n 1 all eigenvalues tend to n 1 2 4

eigenvalue ratios for sturm liouville operators in this paper we prove various optimal bounds for eigenvalue ratios for the sturm liouville equation p x y q x y w x y and certain specializations our results primarily concern the regular case with dirichlet boundary conditions though various extensions and generalizations to other situations are possible our results here extend the result m 1 m2 obtained in a previous paper for the one dimensional schrodinger equation y q x y y on a finite interval with dirichlet boundary conditions and nonnegative potential q 0 in particular we obtain m 1 km2 k where the constants k k satisfy 0 k p x w x k for all x if q 0 lower bounds can also be obtained our methods involve a slight modification of the prufer variable techniques employed in the schrodinger case we also examine the consequences of our recent proof of the payne polya weinberger conjecture in the one dimensional sturm liouville setting finally we compare our general bounds to the detailed analyses of keller and of mahar and willner for the special case of the inhomogeneous stretched string

rearrangement inequalities and applications to isoperimetric problems for eigenvalues let be a bounded c 2 domain in r n where n is any positive integer and let be the euclidean ball centered at 0 and having the same lebesgue measure as consider the operator l div ar vr v on with dirichlet boundary condition where the symmetric matrix

numerical integration of functions with boundary singularities abstract in this paper we deal with the problem of constructing efficient rules for the numerical evaluation of integrals of functions which are very smooth everywhere in the domain of integration except at the boundaries where they possess mild singularities in particular we consider integrals defined on bounded intervals or on triangles integrals of this type appear for example in the numerical solution of singular and weakly singular integral equations by boundary element methods

estimates for the eigenvalues of the bi drifting laplacian on complete metric measure spaces as we know it is very difficult to prove an eigenvalue inequality of ashbaugh cheng ichikawa mametsuka type for the bi drifting laplacian on the bounded domain of the complete metric measure spaces even assumed that the differential operator is bi laplacian on the n dimensional unit euclidean sphere this problem still remains open however a general inequality with respect to the bi drifting laplacian is successfully established under certain conditions in this paper applying the general inequality we prove some eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type on the gaussian shrinking ricci soliton and the n dimensional cigar metric measure spaces cmms for short in particular we obtain two interesting eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type for the cmms with lower dimensional topology

differential inequalities for riesz means and weyl type bounds for eigenvalues we derive differential inequalities and difference inequalities for riesz means of eigenvalues of the dirichlet laplacian r z x k z k

an algorithm for the approximate solution of integral equations of mellin type the cruciform crack problem of elasticity gives rise to an integral equation of the second kind on 0 1 whose kernel has a fixed singularity at 0 0 we introduce a transformation of 0 1 onto itself such that an arbitrary number of derivatives vanish at the end points 0 and 1 if the transformed kernel is dominated near the origin by a mellin kernel then we have given conditions under which the use of a modified euler maclaurin quadrature rule and the nystrom method gives an approximate solution which converges to the exact solution of the original equation the method is illustrated with a numerical example

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

universal inequalities for the eigenvalues of schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schr o dinger operators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schr o dinger operators on homogeneous riemannian spaces and more generally on any riemannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces

on eigenvalues of a system of elliptic equations and of the biharmonic operator abstract let be a bounded domain in an n dimensional euclidean space r n we study eigenvalues of an eigenvalue problem of a system of elliptic equations u grad div u u in u 0 estimates for eigenvalues of the above eigenvalue problem are obtained furthermore we obtain an upper bound on the k 1 th eigenvalue k 1 we also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive ricci curvature

a nystrom method for boundary integral equations in domains with corners we give a convergence and error analysis for a nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic dirichlet problem in plane domains with corners

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

a sharp bound for the ratio of the first two dirichlet eigenvalues of a domain in a hemisphere of ⁿ for a domain q contained in a hemisphere of the n dimensional sphere sn we prove the optimal result a2 ai q a2 al 0 for the ratio of its first two dirichlet eigenvalues where q the symmetric rearrangement of q in sn is a geodesic ball in n having the same n volume as q we also show that a2 a1 for geodesic balls of geodesic radius 01 less than or equal to 7r 2 is an increasing function of 01 which runs between the value jn 2 1 in 2 1 1 2 for 1 0 this is the euclidean value and 2 n l n for 01 7r 2 here jv k denotes the kth positive zero of the bessel function j t this result generalizes the payne polya weinberger conjecture which applies to bounded domains in euclidean space and which we had proved earlier our method makes use of symmetric rearrangement of functions and various technical properties of special functions we also prove that among all domains contained in a hemisphere of sn and having a fixed value of a1 the one with the maximal value of a2 is the geodesic ball of the appropriate radius this is a stronger but slightly less accessible isoperimetric result than that for a2 1i various other results for a1 and a2 of geodesic balls in sn are proved in the course of our work

eigenvalue pinching on convex domains in space forms in this paper we show that the convex domains of the hyperbolic space which are almost extremal for the faber krahn or the payne polya weinberger inequalities are close to geodesic balls our proof is also valid in other space forms and allows us to recover known results in euclidean space and on the sphere

the method of fundamental solutions applied to some inverse eigenproblems in this work we address the application of the method of fundamental solution mfs as a forward solver in some shape optimization problems in two and three dimensional domains it is well known kac s problem that a set of eigenvalues does not determine uniquely the shape of the domain moreover even the existence problem is not well defined due to the ashbaugh benguria inequality although these results constitute counterexamples in the general problem of shape determination from the eigenfrequencies we can address simpler questions in shape determination using the mfs for instance we apply the mfs to build domains that include a specific finite set of eigenvalues or that have an eigenmode that verifies some prescribed conditions as a particular case an eigenmode that defines a certain nodal line

a polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non smooth boundaries the aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1d and 2d weakly singular and singular integral equations the introduction of a smoothing transformation besides smoothing the solution allows also the use of a very simple and efficient collocation method based on chebyshev polynomials of the first kind and their zeros further we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements which are given by weakly singular singular or nearly singular integrals several numerical tests are given to point out the efficiency of the numerical approach we propose copyright 2003 john wiley sons ltd

extremal eigenvalue gaps for the schrodinger operator with dirichlet boundary conditions we consider the problems of minimizing and maximizing the gap between the two lowest dirichlet eigenvalues of the schrodinger operator v x on a bounded domain in rn when the potential v is subjected to a p norm constraint we give characterization theorems for extremizing potentials we prove in particular that a second eigenvalue corresponding to a minimizing potential is always single

universal bounds for eigenvalues of the polyharmonic operators we study eigenvalues of polyharmonic operators on compact riemannian manifolds with boundary possibly empty in particular we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a euclidean space this inequality controls the kth eigenvalue by the lower eigenvalues independently of the particular geometry of the domain our inequality is sharper than the known payne polya weinberg type inequality and also covers the important yang inequality on eigenvalues of the dirichlet laplacian we also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by ashbaugh finally we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

universal bounds for eigenvalues of the biharmonic operator abstract in this paper we study the eigenvalues of the clamped plate problem 2 u u in d u d u d 0 where d is a bounded connected domain in an n dimensional complete minimal submanifold of a unit m sphere s m 1 or of an m dimensional euclidean space r m let 0 1 2 k be the eigenvalues of the above problem we obtain universal bounds on k 1 in terms the first k eigenvalues independent of the domains for example when d is contained in an n dimensional complete minimal submanifold of s m 1 we show that k 1 1 k i 1 k i 1 k n i 1 k k 1 i 1 2 2 n 4 i 1 2 n 2 1 2 i 1 k k 1 i 1 2 4 i 1 2 n 2 1 2 from which one can obtain a more explicit upper bound on k 1 in terms of 1 k see corollary 1 when d is contained in a complete n dimensional minimal submanifold of r m we prove the inequality k 1 1 k i 1 k k 8 n 2 n 2 1 2 1 k i 1 k i k 1 i 1 2 which generalizes the main theorem in cheng and yang 2006 10 that states that the same estimate holds when d is a connected and bounded domain in r n

a collocation method for the generalized airfoil equation for an airfoil with a flap we are concerned with the numerical solution of the generalized airfoil equation for an airfoil with a flap by means of global algebraic polynomial approximants this problem has been considered in a recent paper by monegato and sloan siam j numer anal 34 1997 pp 2288 2305 where the authors proved the stability and convergence of a galerkin method based on high order polynomials they also presented very promising numerical results for the corresponding collocation method but left the theoretical results of stability and convergence as an open problem we have succeeded in proving the stability and convergence properties of this collocation method by slightly modifying it in the neighborhood of the flap the rate of convergence we have derived for our collocation method is very similar to that of the above galerkin method

berezin li yau inequalities on domains on the sphere abstract we prove berezin li yau inequalities for the dirichlet and neumann eigenvalues on domains on the sphere s d 1 a sharp explicit bound for the sums of the neumann eigenvalues is obtained for all dimensions d in the case of s 2 we also obtain sharp lower bounds with correction terms for the vector laplacian and the stokes operator

a discrete collocation method for symm s integral equation on curves with corners this paper is devoted to the approximate solution of the classical first kind boundary integral equation with logarithmic kernel symm s equation on a closed polygonal boundary in 2 we propose a fully discrete method with a trial space of trigonometric polynomials combined with a trapezoidal rule approximation of the integrals before discretization the equation is transformed using a nonlinear mesh grading parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions the convergence results are illustrated with some numerical examples

on the optimization of the principal eigenvalue for single centre point interaction operators in a bounded region we investigate relations between spectral properties of a single centre point interaction hamiltonian describing a particle confined to a bounded domain with a dirichlet boundary and the geometry of for this class of operators krein s formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one d z 1 we use a moving plane analysis to characterize the behaviour of the ground state energy of the hamiltonian with respect to the point interaction position and the shape of in particular we establish some conditions showing how to place the interaction to optimize the principal eigenvalue

commutators spectral trace identities and universal estimates for eigenvalues using simple commutator relations we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self adjoint operator acting in a hilbert space applications involve abstract universal estimates for the eigenvalue gaps as particular examples we present simple proofs of the classical universal estimates for eigenvalues of the dirichlet laplacian as well as of some known and new results for other differential operators and systems we also suggest an extension of the methods to the case of non self adjoint operators

domain independent upper bounds for eigenvalues of elliptic operators let q c ir m be a bounded open set aq its boundary and a the laplacian on rm consider the elliptic differential equation 1 au au in q u 0 on aq it is known that the eigenvalues i of 1 satisfy n m n 2 i l n 1l 1 provided that n l an in this paper we abstract the method used by hile and protter 2 to establish 2 and apply the method to a variety of second order elliptic problems in particular to all constant coefficient problems we then consider a variety of higher order problems and establish an extension of 2 for problem 1 where the laplacian is replaced by a more general operator in a hilbert space

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

numerical solution of the generalized airfoil equation for an airfoil with a flap in this paper we are concerned with the generalized airfoil equation which governs the pressure across an airfoil oscillating in a wind tunnel in particular we examine the case of an airfoil which contains a flap so that the input function has a simple jump at a point x0 1 x0 1 this generates a logarithmic singularity in the solution u at the same point x0 and considerably complicates the problem of the numerical approximation of u by introducing a nonlinear transformation of the independent variable in the integral equation we improve the behavior of the unknown function u through incorporation of the jacobian of the transformation the price is that the kernel of the new integral equation becomes more complicated nevertheless we are able to show that a galerkin method based on high order polynomials is both stable and convergent we also present some promising numerical results for the corresponding collocation method for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the galerkin method

optimal lower bounds for eigenvalues of linear and nonlinear neumann problems in this paper we prove a sharp lower bound for the first non trivial neumann eigenvalue 1 for the p laplace operator p 1 in a lipschitz bounded domain in n our estimate does not require any convexity assumption on and it involves the best isoperimetric constant relative to in a suitable class of convex planar domains our bound turns out to be better than the one provided by the payne weinberger inequality

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

symmetric decreasing rearrangement can be discontinuous suppose f xl x2 0 is a continuously differentiable function supported in the unit disk in the plane its symmetric decreasing rearrangement is the rotationally invariant function f xl x2 whose level sets are circles enclosing the same area as the level sets of f such rearrangement preserves lp norms but decreases convex gradient integrals e g p p 1 p 0 j 1 2 3 is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f the symmetzed functions also converge uniformly the real question is about convergence of the derivatives of the symmetrized functions we announce that the derivatives of the symmetrized functions need not converge strongly e g it can happen that fj f p 0 for every p we further characterize exactly those f s for which convergence is assured and for which it can fail

properties of extremal functions for some nonlinear functionals on dirichlet spaces let 8 be the set of holomorphic functions f on the unit disc d with f o 0 and dirichlet integral 1 7r ffd if 12 not exceeding one and let b be the set of complex valued harmonic functions f on the unit disc with f 0 0 and dirichlet integral 1 2 1 7r ffd lvf12 not exceeding one for a semi continuous function 4 0 oo r define the nonlinear functional a4 on 9 or b by a f f 4j 1f ez0 i do we study the existence and regularity of extremal functions for these functionals as well as the weak semicontinuity properties of the functionals we also state a number of open problems

universal inequalities for eigenvalues of the buckling problem on spherical domains in this paper we study the eigenvalues of the buckling problem on domains in a unit sphere we obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

a second proof of the payne polya weinberger conjecture without using product representations or elaborate comparisons of zeros we prove the two key properties of the bessel function ratiojp 1jp 1 1x jpjp 1x that we used to prove the payne polya weinberger conjecture in these new proofs we use only differential equations and the rayleigh ritz method for estimating lowest eigenvalues the new proofs admit generalization to other related problems where our previous proofs fail

upper and lower bounds for eigenvalues of the clamped plate problem abstract in this paper we study estimates for eigenvalues of the clamped plate problem a sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in cheng and wei 2011 4 is improved

the ppw conjecture in curved spaces abstract in euclidean 1 and hyperbolic 5 space and the round hemisphere 2 geodesic balls maximize the gap 2 1 of dirichlet eigenvalues among domains with fixed 1 we prove an upper bound on 2 1 for domains in manifolds with certain curvature bounds the inequality is sharp on geodesic balls in spaceforms

on sums of graph eigenvalues we use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite simple combinatorial graphs these include estimates for the adjacency matrix of a graph and for both the standard combinatorial laplacian and the renormalized laplacian we also provide upper bounds for sums of squares of eigenvalues of these three matrices among our results we generalize an inequality of fiedler for the extreme eigenvalues of the graph laplacian to a bound on the sums of the smallest or largest k such eigenvalues k n furthermore if lambda j are the eigenvalues of the graph laplacian h delta in increasing order on a finite graph with vertical bar nu vertical bar vertices and vertical bar epsilon vertical bar edges which is isomorphic to a subgraph of the v dimensional infinite cubic lattice then the spectral sums obey a weyl type upper bound a simplification of which reads sigma k 1 j 1 lambda j pi 2 vertical bar epsilon vertical bar 3 k vertical bar nu vertical bar 1 2 v for each k vertical bar nu vertical bar this and related estimates for sigma k 1 j 1 lambda 2 j provide a family of necessary conditions for the embeddability of the graph in a lattice of dimension nu or less c 2014 elsevier inc all rights reserved

universal bounds for eigenvalues of a buckling problem in this paper we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n dimensional euclidean space which is also called a buckling problem we introduce a new method to construct nice trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions thus we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem which was proposed by payne polya and weinberger in 14 and this problem has been mentioned again by ashbaugh in 1

best possible upper and lower bounds for the zeros of the bessel function _ let j k denote the k th positive zero of the bessel function j x in this paper we prove that for 0 and k 1 2 3 ak 21 3 j k ak 21 3 3 20 ak 21 3 1 3 these bounds coincide with the first few terms of the well known asymptotic expansion j k ak 21 3 3 20 ak 21 3 1 3 as k being fixed where ak is the k th negative zero of the airy function ai x and so are best possible

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

on low eigenvalues of the laplacian with mixed boundary conditions abstract by means of the so called symmetrization we study the eigenvalue problem for the laplace operator with mixed boundary conditions we obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of bessel functions

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

fundamental tones of clamped plates in nonpositively curved spaces abstract we study lord rayleigh s problem for clamped plates on an arbitrary n dimensional n 2 cartan hadamard manifold m g with sectional curvature k 2 for some 0 we first prove a mckean type spectral gap estimate i e the fundamental tone of any domain in m g is universally bounded from below by n 1 4 16 4 whenever the cartan hadamard conjecture holds on m g e g in 2 and 3 dimensions due to bol 1941 and kleiner 1992 respectively in 2 and 3 dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates i e the fundamental tone of any domain in m g of volume v 0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature 2 provided that v c n n with c 2 21 031 and c 3 1 721 respectively in particular rayleigh s problem in euclidean spaces resolved by nadirashvili 1992 and ashbaugh and benguria 1995 appears as a limiting case in our setting i e k 0 sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low dimensional hyperbolic spaces are also given the sharpness of our results requires the validity of the cartan hadamard conjecture i e sharp isoperimetric inequality on m g and peculiar properties of the gaussian hypergeometric function both valid only in dimensions 2 and 3 nevertheless some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high dimensions as an application by using the sharp isoperimetric inequality for small clamped hyperbolic discs we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic pde involving the biharmonic laplace beltrami operator

universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds in this paper we consider eigenvalues of the dirichlet biharmonic operator on compact riemannian manifolds with boundary possibly empty and prove a general inequality for them by using this inequality we study eigenvalues of the dirichlet biharmonic operator on compact domains in a euclidean space or a minimal submanifold of it and a unit sphere we obtain universal bounds on the k 1 th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains the estimate for the k 1 th eigenvalue of bounded domains in a euclidean space improves an important inequality obtained recently by cheng and yang

brownian cuboid and its geometric characteristics abstract a class of random cuboid model which is defined brownian cuboid is constructed in this paper based on the structure characteristics of brownian cuboid and the properties of brownian motion geometric characteristics of brownian cuboid are discussed by which the probability estimates of the length of the space diagonal surface area and volume are obtained and moreover the small deviation estimates and some relevant results for the geometric characteristics of brownian cuboid are given additionally we directly derive the upper and lower bounds for the smallest positive zero of the bessel functions of the first kind

universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schrodinger oper ators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schrodinger operators on homogeneous riemannian spaces and more generally on any rie mannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros