existence and multiplicity results for a class of generalized one dimensional p laplacian problem abstract in this paper we study a class of one dimensional p laplacian boundary value problem with generalized sturm liouville boundary conditions some results are obtained for existence of at least three positive solutions by using the related fixed point theory the interesting point is the nonlinear term f which involves the first order derivative explicitly

numerical study of the interface errors of finite difference simulations of seismic waves numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion which is only valid for homogeneous media however there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers the error associated with media that have two layers is called an interface error and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite difference schemes fds for homogeneous media we evaluated a detailed numerical study of the interface error for three staggered grid fds that are commonly used in the simulation of seismic wave propagation we determined that a standard staggered grid scheme ssgs also known as the virieux scheme a rotated staggered grid scheme rsgs and a lebedev scheme ls preserve the second order of convergence at horizontal vertical solid solid interfaces when the medium parameters have been properly modified such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density however for a fluid solid interface aligned with the grid line a second order convergence can only be achieved by an ssgs in addition the presence of a fluid solid interface reduces the order of convergence for the ls and the rsgs to a first order of convergence the presence of inclined interfaces makes high order second and more convergence impossible

construction and analysis of fourth order finite difference schemes for the acoustic wave equation in nonhomogeneous media in this article we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients these schemes are fourth order accurate in space and time in the case of smooth media and are designed to remain stable and optimal for reflection transmission phenomena in the case of discontinuous coefficients together with a detailed mathematical study various numerical results are presented

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

optimized higher order time discretization of second order hyperbolic problems construction and numerical study we investigate explicit higher order time discretizations of linear second order hyperbolic problems we study the even order 2m schemes obtained by the modified equation method we show that the corresponding cfl upper bound for the time step remains bounded when the order of the scheme increases we propose variants of these schemes constructed to optimize the cfl condition the corresponding optimization problem is analyzed in detail the optimal schemes are validated through various numerical results

higher order triangular finite elements with mass lumping for the wave equation in this article we construct new higher order finite element spaces for the approximation of the two dimensional 2d wave equation these elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping these formulas are constructed explicitly error estimates are provided for the corresponding semidiscrete problem finally higher order finite difference time discretizations are proposed and various numerical results are shown

a rearrangement based proof for the existence of extremal functions for the sobolev poincare inequality on bn abstract existence of extremals for 1 p n 1 2 is shown for the sobolev poincare inequality on b n using a mean value zero preserving rearrangement

the best constant for the sobolev trace embedding from w1 1 ω into l1 ω abstract in this paper we study the best constant 1 for the trace map from w 1 1 into l 1 we show that this constant is attained in bv when 1 1 moreover we prove that this constant can be obtained as limit when p 1 of the best constant of w 1 p l p to perform the proofs we will look at neumann problems involving the 1 laplacian 1 u div du du

a symmetry problem related to wirtinger s and poincare s inequality j u 1 1 u p u q among periodic functions in w 1 1 with mean value zero and for p q 1 the minimizers are scaled in such a way that their maximum is 1 and their minimum is m 0 1 they raise the question if m 1 since this implies symmetries for the minimizers their answer is positive for q 2p and negative for q 1 practically the same question is raised in 2 and 3 if m 1 minimizers must be odd or symmetric in the sense that u x u x it was shown in 3 that for p q 2 the minimizer is symmetric and that for p 2 q it is not symmetric moreover in 2 one can find that the minimizer is symmetric for q 2p and nonsymmetric for for q 4p 1 the nonsymmetry result was derived by egorov in the following way restricted to the following set a

sharp hardy inequalities in the half space with trace remainder term abstract in this paper we deal with a class of inequalities which interpolate the kato inequality and the hardy inequality in the half space starting from the classical hardy s inequality in the half space r n r n 1 0 we show that if we replace the optimal constant n 2 2 4 with a smaller one 2 2 4 2 n then we can add an extra trace term similar to that one that appears in the kato inequality the constant in the trace remainder term is optimal and it tends to zero when goes to n while it is equal to the optimal constant in kato s inequality when 2 the approach is based on a very classical method of calculus of variation due to weierstrass and developed by hilbert that usually is considered to prove that the solutions of the euler lagrange equation associated to a functional are in fact extremals in this paper we will show how this method is well suited also to functionals that have no extremals

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

domain perturbation for elliptic equations subject to robin boundary conditions abstract we prove domain perturbation theorems for linear and nonlinear elliptic equations under robin boundary conditions the theory allows very singular perturbation of domains in particular it includes cutting holes parts degenerating to a set of measure zero such as the dumbbell problem or wildly oscillating boundaries in the last case we show that the limiting problem is the dirichlet problem

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

symmetric solutions of a multi point boundary value problem abstract we apply the fixed point theorem of avery and peterson to the nonlinear second order multi point boundary value problem u t a t f t u t u t t 0 1 u 0 i 1 n i u i u 1 t u t t 0 1 where 0 1 2 n 1 2 i 0 for i 1 n with i 1 n i 1 n 2 we show that under the appropriate growth conditions on the inhomogeneous term symmetric about t 1 2 the problem has triple symmetric solutions

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

an efficient high order algorithm for acoustic scattering from penetrable thin structures in three dimensions this paper presents a high order accelerated algorithm for the solution of the integral equation formulation of volumetric scattering problems the scheme is particularly well suited to the analysis of thin structures as they arise in certain applications e g material coatings in addition it is also designed to be used in conjunction with existing low order fft based codes to upgrade their order of accuracy through a suitable treatment of material interfaces the high order convergence of the new procedure is attained through a combination of changes of parametric variables to resolve the singularities of the green function and partitions of unity to allow for a simple implementation of spectrally accurate quadratures away from singular points accelerated evaluations of the interaction between degrees of freedom on the other hand are accomplished by incorporating two face equivalent source approximations on cartesian grids a detailed account of the main algorithmic components of the scheme are presented together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

solutions of the cheeger problem via torsion functions abstract the cheeger problem for a bounded domain r n n 1 consists in minimizing the quotients e e among all smooth subdomains e and the cheeger constant h is the minimum of these quotients let p c 1 be the p torsion function that is the solution of torsional creep problem p p 1 in p 0 on where p u div u p 2 u is the p laplacian operator p 1 the paper emphasizes the connection between these problems we prove that lim p 1 p l 1 p h lim p 1 p l 1 1 p moreover we deduce the relation lim p 1 p l 1 c n lim p 1 p l where c n is a constant depending only of n and h explicitely given in the paper an eigenfunction u bv l of the dirichlet 1 laplacian is obtained as the strong l 1 limit as p 1 of a subsequence of the family p p l 1 p 1 almost all t level sets e t of u are cheeger sets and our estimates of u on the cheeger set e 0 yield b 1 h b 1 n e 0 h n where b 1 is the unit ball in r n for convex we obtain u e 0 1 e 0

dirichlet problems on varying domains abstract the aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators examples are given to illustrate that most results are optimal

numerical diffraction by a uniform grid in this report we analyze the influence of a spatial discretization on the propagation of a wave generated by a harmonic point source we consider finite element and finite difference schemes on a uniform grid for the wave equation we study the asymptotic behavior in space of the elementary solution of the associated helmholtz equation this study permits us to define the notions of asymptotic wave fronts a w f and asymptotic amplitude diagrams a a d we point out some results about the anisotropy and the dispersion of numerical scheme which complete those obtained from the more classical plane wave analysis see r m alford r kelly and d m boore accuracy of finite modeling of the acoustic wave equation biophysics 6 1974 pp 834 842 a bamberger g chavent and p lailly etude de schemas numeriques des equations de l elastodynamique lineaire rapport inria no 41 october 1980 t belytschko and r mullen dispersion analysis of finite element semi discretizations of the two dimen

a sharp trace inequality for functions of bounded variation in the ball the best constant in a mean value trace inequality for functions of bounded variation on admissible domains n is shown to agree with an isoperimetric constant associated with the existence and form of extremals is also discussed this result is exploited to compute the best constant in the relevant trace inequality when is a ball the existence and the form of extremals in this special case turn out to depend on the dimension n in particular the best constant is not achieved when is a disc in 2

efficient and accurate numerical simulation of acoustic wave propagation in a 2d heterogeneous media abstract in this paper a compact fourth order finite difference scheme is derived to solve the 2d acoustic wave equation in heterogenous media the pade approximation is used to obtain fourth order accuracy in both temporal and spatial dimensions and the alternating direction implicit adi technique is used to reduce the computational cost due to the non constant wave velocity the conventional adi method is hard to implement as the algebraic manipulation cannot be used here a novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth order accuracy in time and space the fourth order convergence order was firstly proved by theoretical error analysis then was confirmed by numerical examples it was shown that the proposed method is conditionally stable with a courant friedrichs lewy cfl condition that is comparable to other existing finite difference schemes several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm

sobolev embeddings into spaces of campanato morrey and holder type abstract necessary and sufficient conditions on a rearrangement invariant banach function space x q on a cube q in r n n 2 are given for the corresponding sobolev space w1x q to be continuously embedded into generalized campanato morrey or holder spaces the optimal such r i spaces x q are found as a by product sharp inclusion relations are proved among campanato morrey and holder type spaces

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

a theoretical analysis of a new second order finite volume approximation based on a low order scheme using general admissible spatial meshes for the one dimensional wave equation abstract this paper details our note 6 and it is an extension of our previous works 4 5 which dealt with first order both in time and space and second order time accurate second order in time and first order in space implicit finite volume schemes for second order hyperbolic equations with dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in 14 we aim in this work and some forthcoming studies to get higher order both in time and space finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in 14 on low order schemes from which the matrices used to compute the discrete solutions are sparse we focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in 4 the implicit finite volume scheme approximating the one dimensional wave equation we consider hereafter referred to as the basic finite volume scheme yields linear systems to be solved successively the matrices involved in these linear systems are tridiagonal symmetric and definite positive the finite volume approximate solution of the basic finite volume scheme is of order h k where h resp k is the mesh size of the spatial resp time discretization we construct a new finite volume approximation of order h k 2 in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives this new high order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected the construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution the computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices the manner by which this new high order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above these high order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of 13 without any condition to reach the above results a theoretical framework is developed and some numerical examples supporting the theory are presented some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions in particular a new and useful a prior estimate for a suitable discrete problem is developed and proved the proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems a new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

robin boundary value problems on arbitrary domains we develop a theory of generalised solutions for elliptic boundary value problems subject to robin boundary conditions on arbitrary domains which resembles in many ways that of the dirichlet problem in particular we establish lp lq estimates which turn out to be the best possible in that framework we also discuss consequences to the spectrum of robin boundary value problems finally we apply the theory to parabolic equations

a new high accuracy locally one dimensional scheme for the wave equation in this paper a new locally one dimensional lod scheme with error of o dt 4 h 4 for the two dimensional wave equation is presented the new scheme is four layer in time and three layer in space one main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step the stability and dispersion analysis of the new scheme are given the computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed which makes the scheme suitable for performing practical simulation in wave propagation modeling furthermore a comparison of our new scheme and the traditional finite difference scheme is given which shows the superiority of our new method

multi point boundary value problems of second order differential equations i abstract we study the nonlinear boundary value problem with nonhomogeneous multi point boundary condition u f t u u 0 t 0 1 u 0 i 1 m a i u t i 1 u 1 i 1 m b i u t i 2 sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relations using one of the results under some assumptions we obtain explicit ranges of values of 1 and 2 with which the problem has a solution has a positive solution and has no solution respectively furthermore we prove that the whole plane for 1 and 2 can be divided into two disjoint connected regions e and n such that the problem has a solution for 1 2 e and has no solution for 1 2 n

persistence of the bifurcation structure for a semilinear elliptic problem on thin domains we study a semilinear elliptic problem on thin domains with a bifurcation parameter it is shown that the set of solutions is upper semicontinuous as the thickness of a domain tends to 0 and that solution branches including bifurcation points persist near those of a one dimensional limiting equation

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method

multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one dimensional p laplacian abstract in this paper we consider the multipoint boundary value problem for the one dimensional p laplacian p u t q t f t u t u t 0 t 0 1 subject to the boundary conditions u 0 i 1 n i u i u 1 i 1 n i u i where p s s p 2 s p 1 i 0 0 i 1 n i 1 0 1 2 n 1 2 i i 1 i 1 2 n applying a fixed point theorem of functional type in a cone we study the existence of at least three symmetric positive solutions to the above boundary value problem the interesting point is that the nonlinear term f contains the first order derivative explicitly

l bounds of steklov eigenfunctions and spectrum stability under domain variation abstract we give a practical tool to control the l norm of the steklov eigenfunctions in a lipschitz domain in terms of the norm of the bv trace operator the norm of this operator has the advantage to be characterized by purely geometric quantities as a consequence we give a spectral stability result for the steklov eigenproblem under geometric domain perturbations and several examples where stability occurs in particular we deal with geometric domains which are not equi lipschitz like vanishing holes merging sets approximations of inner peaks

a staggered time integrator for the linear acoustic wave equation using the jacobi anger expansion abstract in this study we introduce a staggered time integrator with the fourier pseudospectral method to solve the first order linear wave equation which is accelerated by using the jacobi anger expansion the proposed method can reduce the computational cost by approximately half compared to the scheme whose temporal order of accuracy is extended by the lax wendroff method under the equivalent modeling conditions such as time step length grid interval and maximum wave propagation speed this is because the jacobi anger expansion can effectively approximate the sinusoidal function which in our case is the sine function in the wavenumber domain based on the wavenumber domain analysis of the proposed method a strategy to optimally design the simulation parameters to maintain a preset level of accuracy is also introduced according to the strategy as the time step length increases not only the computational cost of the simulation is reduced but also the accuracy of the numerical solution is improved numerical simulations are also performed by using both homogeneous and heterogeneous models to validate the introduced strategy which supports the practicality of the proposed method

the longest shortest fence and sharp poincare sobolev inequalities we prove a long standing conjecture concerning the fencing problem in the plane among planar convex sets of given area the disc and only the disc maximizes the length of the shortest area bisecting curve although it may look intuitive the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so called auerbach triangle

time compact high order difference methods for wave propagation we construct one step explicit difference methods for solution of wave propagation problems with fourth order accuracy in both space and time by using a principle that can be generalized to arbitrary order we use the first order system form and a staggered grid the fourth order accuracy in time is obtained by transferring time derivatives in the truncation error to space derivatives discontinuous coefficients corresponding to interfaces between different materials are considered as a special case of variable coefficients and the method is applied across the discontinuities the accuracy is much improved compared to second order methods even for this type of problem a certain norm is shown to be conserved ensuring good accuracy even for long time integration

local and global uniform convergence for elliptic problems on varying domains abstract the aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with dirichlet boundary conditions on varying domains we assume that the limit domain be stable in the sense of keldys amer math soc transl 51 1966 1 73 we further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain and only require l 2 convergence outside as a consequence uniform and l 2 convergence are the same in the trivial case of homogenisation of a perforated domain we are also able to deal with certain cracking domains

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

optimal existence criteria for symmetric positive solutions to a three point boundary value problem abstract in this paper we shall consider the following second order three point boundary value problem u t a t f t u t 0 0 t 1 u 0 u 1 u where 0 1 0 1 2 a l 1 0 1 is nonnegative and symmetric on 0 1 f 0 1 0 0 satisfies caratheodory conditions and f u is symmetric on 0 1 for all u 0 by using fixed point index theorems we get some optimal existence criteria for the existence of one or two symmetric solutions which involve the principal eigenvalue of a related linear operator

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

a variational approach to the isoperimetric inequality for the robin eigenvalue problem the isoperimetric inequality for the first eigenvalue of the laplace operator with robin boundary conditions was recently proved by daners in the context of lipschitz sets this paper introduces a new approach to the isoperimetric inequality based on the theory of special functions of bounded variation sbv we extend the notion of the first eigenvalue 1 for general domains with finite volume possibly unbounded and with irregular boundary and we prove that the balls are the unique minimizers of 1 among domains with prescribed volume

an efficient fourth order low dispersive finite difference scheme for a 2 d acoustic wave equation in this paper we propose an efficient fourth order compact finite difference scheme with low numerical dispersion to solve the two dimensional acoustic wave equation combined with the alternating direction implicit adi technique and pade approximation the standard second order finite difference scheme can be improved to fourth order and solved as a sequence of one dimensional problems with high computational efficiency however such compact higher order methods suffer from high numerical dispersion to suppress numerical dispersion the compact and non compact stages are interlinked to produce a hybrid scheme in which the compact stage is based on pade approximation in both y and temporal dimensions while the non compact stage is based on pade approximation in y dimension only stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the courant friedrichs lewy cfl condition the dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact adi scheme and the higher order locally one dimensional lod scheme three numerical examples are solved to demonstrate the accuracy and efficiency of the new method

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

a compact high order alternating direction implicit method for three dimensional acoustic wave equation with variable coefficient abstract efficient and accurate numerical simulation of seismic wave propagation is important in various geophysical applications such as seismic full waveform inversion fwi problem however due to the large size of the physical domain and requirement on low numerical dispersion many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media despite the great efforts that have been devoted during the past decades it still remains a challenging task in the development of efficient and accurate finite difference method for multi dimensional acoustic wave equation with variable velocity in this paper we proposed a pade approximation based finite difference scheme for solving the acoustic wave equation in three dimensional heterogeneous media the new method is obtained by combining the pade approximation and a novel algebraic manipulation the efficiency of the new algorithm is further improved through the alternative directional implicit adi method the stability of the new algorithm has been theoretically proved by energy method the new method is conditionally stable with a better courant friedrichs lewy condition cfl condition which has been verified numerically extensive numerical examples have been solved which demonstrated that the new method is accurate efficient and stable

torsional rigidity of minimal submanifolds we prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds p m in ambient riemannian manifolds n n with a pole p the upper bounds are given in terms of the torsional rigidities of corresponding schwarz symmetrizations of the domains in warped product model spaces our main results are obtained via previously established isoperimetric inequalities which are here extended to hold for this more general setting based on warped product comparison spaces we also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for brownian motion at infinity is finite

schrodinger operators with negative potentials and lane emden densities abstract we consider the schrodinger operator v for negative potentials v on open sets with positive first eigenvalue of the dirichlet laplacian we show that the spectrum of v is positive provided that v is greater than a negative multiple of the logarithmic gradient of the solution to the lane emden equation u u q 1 for some 1 q 2 in this case the ground state energy of v is greater than the first eigenvalue of the dirichlet laplacian up to an explicit multiplicative factor this is achieved by means of suitable hardy type inequalities that we prove in this paper

on the torsion function with robin or dirichlet boundary conditions for p2 1 1 and b2 0 1 the p torsion function with robin boundary conditions associated to an arbitrary open set r m satises for

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

stability of an explicit high order spectral element method for acoustics in heterogeneous media based on local element stability criteria this paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high order spectral elements the global stability criterion is taken as a minimum over local element stability criteria obtained through the solution of element borne eigenvalue problems first an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and or rapidly fluctuating medium using asymptotic analysis this criterion is only dependent upon the maximum velocity at the vertices of the mesh elements and not on the velocity at the interior nodes of the high order elements second in a more general setting bounds are derived using statistics of the coefficients of the elemental dispersion matrices different bounds are presented discussed and compared several numerical experiments show the accuracy of the proposed criteria in one dimensional test cases as well as in more realistic large scale 3d problems

on the dispersion stability and accuracy of a compact higher order finite difference scheme for 3d acoustic wave equation abstract in this paper we propose a compact fourth order finite difference scheme with low numerical dispersion to solve the 3d acoustic wave equation pade approximation has been used to obtain fourth order accuracy in both temporal and spatial dimensions while the alternating direction implicit adi technique has been used to reduce the computational cost error analysis has been conducted to show the fourth order accuracy which has been confirmed by a numerical example we have also shown that the proposed method is conditionally stable with a courant friedrichs lewy cfl condition that is comparable to other existing finite difference schemes due to the higher order accuracy the new method is found effective in suppressing numerical dispersion

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

the shape of extremal functions for poincare sobolev type inequalities in a ball we study extremal functions for a family of poincare sobolev type inequalities these functions minimize for subcritical or critical p 2 p 2 the quotient u 2 u p u 2 u p among all u h 1 b 0 u h 1 b 0 with b u 0 b u 0 here b is the unit ball in rn r n we show that the minimizers are axially symmetric with respect to a line passing through the origin we also show that they are strictly monotone in the direction of this line in particular they take their maximum and minimum precisely at two antipodal points on the boundary of b we also prove that for p close to 2 minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis and that once the symmetry axis is fixed they are unique up to multiplication by a constant in space dimension two we prove that minimizers are not antisymmetric for large p

an isoperimetric inequality and the first steklov eigenvalue let mn g be a compact riemannian manifold with boundary in this paper we give upper and lower estimates for the first nonzero steklov eigenvalue 0inm 1 on m where 1 is a positive real number the estimate from below is for a star shaped domain on a manifold whose ricci curvature is bounded from below the upper estimate is for a convex manifold with nonnegative ricci curvature and is given in terms of the first nonzero eigenvalue for the laplacian on the boundary we prove a comparison theorem for simply connected domains in a simply connected manifold we exhibit annuli domains for which the comparison theorem fails to be true in j f escobar j funct anal 60 1997 544 556 we introduced the isoperimetric constant i m defined asi m inf mvol min vol 1 vol 2 where 1 m is a nonempty domain with boundary in the manifold m 2 m 1 and int m where int m is the interior of m we proved a cheeger s type inequality for 1 using the constant i m in this paper we give upper and lower estimates for the constant i in terms of isoperimetric constants of the boundary of m

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

monotone positive solutions of second order multi point boundary value problems abstract in this paper by using guo krasnosel skii fixed point theorem and schauder s fixed point theorem we establish sufficient conditions for the existence and nonexistence of monotone increasing positive solution for a class of second order differential equation with multi point nonhomogeneous boundary conditions

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

existence and multiplicity of symmetric positive solutions for three point boundary value problem in this paper we are concerned with the existence and multiplicity of symmetric positive solutions for the following second order three point boundary value problem u t a t f t u t 0 0 t 1 u t u 1 t u 0 u 1 u 1 2 where a 0 1 0 is symmetric on 0 1 and maybe singular at t 0 and t 1 f 0 1 0 0 is continuous and f u is symmetric on 0 1 for all u 0 growth conditions are imposed on f which yield the existence of at least one or at least two positive solutions our proof based on krasnoselskii s fixed point theorem in a cone