sharp generalizations of the multilinear bohnenblust hille inequality we prove that the multilinear bohnenblust hille inequality is a particular case of a quite general family of optimal inequalities

on the eigenvalue problem involving the weighted p laplacian in radially symmetric domains abstract we investigate the following eigenvalue problem div l x u p 2 u k x u p 2 u in a r 1 r 2 u 0 on a r 1 r 2 where a r 1 r 2 x r n r 1 x r 2 0 r 1 r 2 0 is a parameter the weights l and k are measurable with l positive a e in a r 1 r 2 and k possibly sign changing in a r 1 r 2 we prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions the asymptotic estimates for u x and u x as x r 1 or r 2 are also investigated

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

time exponential splitting technique for the klein gordon equation with hagstrom warburton high order absorbing boundary conditions klein gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases numerical computation is reduced to a finite domain by using the hagstrom warburton h w high order absorbing boundary conditions abcs time integration is made by means of exponential splitting schemes that are efficient and easy to implement in this way it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided the influence of the dispersion coefficient on the error is also studied

there exist multilinear bohnenblust hille constants cn n 1 with limn cn 1 cn 0 abstract the n linear bohnenblust hille inequality asserts that there is a constant c n 1 such that the l 2 n n 1 norm of u e i 1 e i n i 1 i n 1 n is bounded above by c n times the supremum norm of u for any n linear form u c n c n c and n n the same holds for real scalars we prove what we call fundamental lemma which brings new information on the optimal constants k n n 1 for both real and complex scalars for instance k n 1 k n 0 87 n 0 473 for infinitely many n s for complex scalars we give a formula of surprisingly low growth in which e and the famous euler mascheroni constant appear k n 1 4 1 e 2 1 2 j 1 n 1 j log 2 e 2 1 2 1 n 2 we study the interplay between the kahane salem zygmund and the bohnenblust hille polynomial and multilinear inequalities and provide estimates for bohnenblust hille type inequality constants for any exponent q 2 n n 1

a subexponential vector valued bohnenblust hille type inequality abstract bayart pellegrino and seoane recently proved that the polynomial bohnenblust hille inequality for complex scalars is subexponential we show that a vector valued polynomial bohnenblust hille inequality on complex banach lattices is also subexponential for some special cases our main result recovers the best known constants of the classical polynomial inequality provided in 4

a d n alternating algorithm for exterior 3 d poisson problem with prolate spheroid boundary in this paper a d n alternating algorithm based on the natural boundary reduction nbr is discussed to solve exterior three dimensional 3 d poisson problem with prolate spheroid artificial boundary by the principle of the natural boundary reduction we obtain the natural integral equation on prolate spheroid artificial boundary suggest a d n alternating algorithm and analysis its convergence of the algorithm finally some numerical examples are presented to show the effectiveness of this method

a stable high order perturbation of surfaces method for numerical simulation of diffraction problems in triply layered media the accurate numerical simulation of linear waves interacting with periodic layered media is a crucial capability in engineering applications in this contribution we study the stable and high order accurate numerical simulation of the interaction of linear time harmonic waves with a periodic triply layered medium with irregular interfaces in contrast with volumetric approaches high order perturbation of surfaces hops algorithms are inexpensive interfacial methods which rapidly and recursively estimate scattering returns by perturbation of the interface shape in comparison with boundary integral element methods the stable hops algorithm we describe here does not require specialized quadrature rules periodization strategies or the solution of dense non symmetric positive definite linear systems in addition the algorithm is provably stable as opposed to other classical hops approaches with numerical experiments we show the remarkable efficiency fidelity and accuracy one can achieve with an implementation of this algorithm

the bohr radius of the n dimensional polydisk is equivalent to sqrt frac log n n we show that the bohr radius of the polydisk mathbb d n behaves asymptotically as sqrt log n n our argument is based on a new interpolative approach to the bohnenblust hille inequalities which allows us to prove that the polynomial bohnenblust hille inequality is subexponential

corrigendum to existence and some properties of solutions for degenerate elliptic equations with exponent variable nonlinear anal 98 2014 146 164 abstract we restate theorem 4 2 and lemma 4 3 in 1 a little differently and give a proof of lemma 4 3 and replace corresponding parts in the proof of theorem 4 2

a general approach to extend berenger s absorbing boundary condition to anisotropic and dispersive media previous studies have focused on the extension of the berenger s 1994 absorbing boundary condition abc the perfectly matched layer pml to anisotropic and dispersive media we describe how the pml can be extended for anisotropic and or dispersive media or even bianisotropic media with little analytical effort in a simple and general conceptual setting by using an analytical continuation of the coordinate space of the maxwell s equations to a complex variables domain using this approach there is no conceptual distinction between the derivation of pml for more complex media and the derivation for isotropic dispersionless media

an optimized hybrid convolutional perfectly matched layer for efficient absorption of electromagnetic waves abstract a hybrid technique is studied in order to improve the performance of convolutional perfectly matched layer cpml in the finite difference time domain fdtd medium this technique combines the first order of higdon s annihilation equation as absorbing boundary condition abc with cpml to vanish the perfect electric conductor pec effects at the end of the cpml region an optimization algorithm is required to find optimum parameters of the proposed absorber in this investigation the particle swarm optimization pso is utilized with two separate objective functions in order to minimize the average and peak value of relative error using a standard test the overall performance of the proposed absorber is compared to the original cpml the results clearly illustrate this method provides approximately 10 db enhancements in cpml absorption error the performance memory and time requirement of the novel absorber hybrid cpml h cpml was analyzed during 2d and 3d tests and compared to most reported pmls the h cpml requirement of computer resources is similar to cpml and can simply be implemented to truncate fdtd domains furthermore an optimized set of parameters are provided to generalize the hybrid method

the domain of analyticity of dirichlet neumann operators dirichlet neumann operators arise in many applications in the sciences and this has inspired a number of studies on their analytical properties in this paper we further investigate the analyticity properties of dirichlet neumann operators as functions of the boundary shape in particular we study the size of the disc of convergence of their taylor series representation for this we use a complexification technique which requires a novel reformulation of the problem coupled with methods for systems of elliptic partial differential equations numerical results to illustrate our theoretical conclusions are presented

coordinatewise multiple summing operators in banach spaces we invent the new notion of coordinatewise multiple summing operators in banach spaces and use it to study various vector valued extensions of the well know bohnenblust hille inequality which originally extended littlewood s 4 3 inequality our results have application on the summability of monomial coefficients of m homogeneous polynomials p l lp as well as for the convergence theory of products of vector valued dirichlet series

high order boundary conditions and finite elements for infinite domains a finite element method for the solution of linear elliptic problems in infinite domains is proposed the two dimensional laplace helmholtz and modified helmholtz equations outside an obstacle and in a semi infinite strip are considered in detail in the proposed method an artificial boundary b is first introduced to make the computational domain finite then the exact nonlocal dirichlet to neumann dtn boundary condition is derived on b this condition is localized and a sequence of local boundary conditions on b of increasing order is obtained the problem in with a localized dtn boundary condition on b is then solved using the finite element method the numerical stability of the scheme is discussed a hierarchy of special conforming finite elements is developed and used in the layer adjacent to b in conjunction with the local high order boundary condition applied on b an error analysis is given for both nonlocal and local boundary conditions numerical experiments are presented to demonstrate the performance of the method

factorization of absolutely continuous polynomials abstract in this paper we study the ideal of dominated p continuous polynomials that extend the nowadays well known ideal of p dominated polynomials to the more general setting of the interpolated ideals of polynomials we give the polynomial version of pietsch s factorization theorem for this new ideal although based on botelho et al in press 16 our factorization theorem requires new techniques inspired in the theory of banach lattices

new upper bounds for the constants in the bohnenblust hille inequality a classical inequality due to bohnenblust and hille states that for every positive integer m there is a constant c m 0 so that sigma n i1 im 1 vertical bar u e i1 e im vertical bar 2m m 1 m 1 2m c where c m m m 1 2m 2 m 1 2 the value of c m was improved to c m 2 m 1 2 by s kaijser and more recently h queffelec and a defant and p sevilla peris remarked that c m 2 root pi m 1 also works the bohnenblust hille inequality also holds for real banach spaces with the constants c m 2 m 1 2 in this note we show that a recent new proof of the bohnenblust hille inequality due to defant popa and schwarting provides in fact quite better estimates for c m for all values of m is an element of n in particular we will also show that for real scalars if m is even with 2 m 24 then c r m 2 1 2 c r m 2 we will mainly work on a paper by defant popa and schwarting giving some remarks about their work and explaining how to numerically improve the previously mentioned constants

improving the closed formula for subpolynomial constants in the multilinear bohnenblust hille inequalities abstract for k r or c the bohnenblust hille inequality asserts that there exists a sequence of scalars c k m m 1 such that i 1 i m 1 n u e i 1 e i m 2 m m 1 m 1 2 m c k m sup z 1 z m d n tu z 1 z m for all m linear forms u k n k n k and every positive integer n where e i i 1 n denotes the canonical basis of k n and d n represents the open unit polydisc in k n very recently 2012 it was shown that there exist constants c k m m 1 with subpolynomial growth satisfying this inequality however these constants were obtained via a complicated recursive formula we improve the best known closed non recursive approximation for these constants

perfectly matched layers for transient elastodynamics of unbounded domains one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies in a recent work computer methods in applied mechanics and engineering 2003 192 1337 1375 the authors presented inter alia time harmonic governing equations of pmls for anti plane and for plane strain motion of visco elastic media this paper presents a corresponding time domain displacement based governing equations of these pmls and b displacement based finite element implementations of these equations suitable for direct transient analysis the finite element implementation of the anti plane pml is found to be symmetric whereas that of the plane strain pml is not numerical results are presented for the anti plane motion of a semi infinite layer on a rigid base and for the classical soil structure interaction problems of a rigid strip footing on i a half plane ii a layer on a half plane and iii a layer on a rigid base these results demonstrate the high accuracy achievable by pml models even with small bounded domains

weak boundary conditions for wave propagation problems in confined domains formulation and implementation using a variational multiscale method abstract we propose a new approach to the enforcement of dirichlet neumann or robin boundary conditions in finite element computations of wave propagation problems the key idea is to enforce the boundary conditions weakly as part of the variational formulation due to the hyperbolic structure of the problem considered the variational formulation does not require any penalty parameters in contrast with what typically happens in elliptic or advection diffusion parabolic problems this article presents the implementation of the proposed boundary condition framework using a variational multiscale method for the wave equation in mixed form we conclude with an extensive set of tests to validate the robustness and accuracy of the proposed approach

perfectly matched layers for flexural waves in kirchhof love plates abstract we propose perfectly matched layers pmls for flexural waves in plate structures the analytical model is based on transformation optics techniques applied to the biharmonic fourth order partial differential equation describing flexural vibrations in kirchhoff love plates we show that perfect boundary conditions are not an optimal solution since they depend on the incident waves the full analytical form of pmls and zero reflection conditions at the boundary between homogeneous and pml domains are given the implementation in a finite element fem code is described and an eigenfrequency analysis is given as a possible methodology to check the implementation itself a measure of the performances of the pmls is introduced and the effects of element discretization boundary conditions frequency dimension of the pml amount of transformation and dissipation are detailed it is shown that the model gives excellent results also when the applied load approaches the pml domain

a new multilinear insight on littlewood s 4 3 inequality abstract we unify littlewood s classical 4 3 inequality a forerunner of grothendieck s inequality together with its m linear extension due to bohnenblust and hille which originally settled bohr s absolute convergence problem for dirichlet series with a scale of inequalties of bennett and carl in l p spaces which are of fundamental importance in the theory of eigenvalue distribution of power compact operators as an application we give estimates for the monomial coefficients of homogeneous l p valued polynomials on c 0

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

the asymptotic growth of the constants in the bohnenblust hille inequality is optimal the search of sharp estimates for the constants in the bohnenblust hille inequality besides its challenging nature has quite important applications in different fields of mathematics and physics for homogeneous polynomials it was recently shown that the bohnenblust hille inequality for complex scalars is hypercontractive this result interesting by itself has found direct striking applications in the solution of several important problems for multilinear mappings precise information on the asymptotic behavior of the constants of the bohnenblust hille inequality is of particular importance for applications in quantum information theory and multipartite bell inequalities in this paper using elementary tools we prove a quite surprising result the asymptotic growth of the constants in the multilinear bohnenblust hille inequality is optimal besides its intrinsic mathematical interest and potential applications to different areas the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years such as for instance the multilinear version of the famous grothendieck theorem for absolutely summing operators always present constants c m s growing at an exponential rate of certain power of m

three dimensional solid to beam transition elements for structural dynamics analysis complex structural components such as those encountered in many industrial applications may generally be considered as being composed of shell or beam like portions linked to three dimensional solid continua when discretized into finite elements these structures present geometrical and mathematical difficulties at the connections between the different element types since the nodal degrees of freedom allocated to the solid shell and beam elements are incompatible with each other the development of specific and reliable transition finite elements is thus of outstanding practical importance this paper presents efficient c0 compatible transition elements with a variable number of nodes for modelling solid to beam junctions based upon the standard isoparametric solid and beam formulations the current approach includes the properties of both solids and beams verifies the basic continuity smoothness and completeness criteria inherent in the finite element convergence requirements and avoids the shear locking phenomenon typical of c0 elements by using a strain projection method several numerical examples which compare this formulation to analytical and experimental solutions are provided in order to show the applicability and efficiency of this approach

absolutely summing multilinear operators via interpolation abstract we use an interpolative technique from 1 to introduce the notion of multiple n separately summing operators our approach extends and unifies some recent results for instance we recover the best known estimates of the multilinear bohnenblust hille constants due to f bayart d pellegrino and j seoane sepulveda more precisely as a consequence of our main result for 1 t 2 and m 1 we prove that i 1 i m 1 u e i 1 e i m 2 t m 2 m 1 t 2 m 1 t 2 t m j 2 m 2 2 t j t 2 t 2 t j 2 2 2 t 2 j t u for all complex m linear forms u c 0 c 0 c

efficient enforcement of far field boundary conditions in the transformed field expansions method the method of transformed field expansions tfe has been demonstrated to be a robust and highly accurate numerical scheme for simulating solutions of boundary value and free boundary problems from the sciences and engineering as a boundary perturbation method it builds highly accurate solutions based upon exact solutions in a simple canonical geometry and corrects these via taylor series to fit the actual geometry at hand the tfe method has significantly enhanced stability properties when compared with other boundary perturbation approaches however this comes at the cost of requiring a full volumetric discretization as opposed the surface formulation that other methods can realize in this paper we outline two techniques for ameliorating this shortcoming first by employing a legendre spectral element method to implement efficient graded meshes and second by utilizing an artificial boundary with a transparent boundary condition placed quite close to the boundary of the domain in this contribution we focus on the specific problem of simulating the dirichlet neumann operator associated to laplace s equation on a periodic cell which arises in the water wave problem while the details of our results are specific to this problem the general conclusions are valid for the wider class of problems to which the tfe method can be applied for each technique we discuss implementation details and display numerical results which support the conclusion that each of these techniques can greatly reduce the computational cost of using the tfe method

on the upper bounds for the constants of the hardy littlewood inequality abstract the best known upper estimates for the constants of the hardy littlewood inequality for m linear forms on l p spaces are of the form 2 m 1 we present better estimates which depend on p and m an interesting consequence is that if p m 2 then the constants have a subpolynomial growth as m tends to infinity

antimaximum principle in exterior domains abstract we consider the antimaximum principle for the p laplacian in the exterior domain p u k x u p 2 u h x in b 1 c u 0 on b 1 where p is the p laplace operator with p 1 is the spectral parameter and b 1 c is the exterior of the closed unit ball in r n with n 1 the function h is assumed to be nonnegative and nonzero however the weight function k is allowed to change its sign for k in a certain weighted lebesgue space we prove that the antimaximum principle holds locally a global antimaximum principle is obtained for h with compact support for a compactly supported k with n 1 and p 2 we provide a necessary and sufficient condition on h for the global antimaximum principle in the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

anisotropic semipositone quasilinear problems abstract we present the existence of positive solutions for a degenerate or singular quasilinear equation in a ball or in the entire space r n as differential operator we consider a weighted p laplacian with a continuous coefficient satisfying a mild integrability condition towards the boundary of the ball or at infinity in r n our main result applies when the nonlinearities f u have p 1 sublinear growth at infinity and singular semipositone structure at the origin

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

non conforming galerkin finite element methods for local absorbing boundary conditions of higher order a new non conforming finite element discretization methodology for second order elliptic partial differential equations involving higher order local absorbing boundary conditions in 2d and 3d is proposed the novelty of the approach lies in the application of c 0 continuous finite element spaces which is the standard discretization of second order operators to the discretization of boundary differential operators of order four and higher for each of these boundary operators additional terms appear on the boundary nodes in 2d and on the boundary edges in 3d similarly to interior penalty discontinuous galerkin methods which leads to a stable and consistent formulation in this way no auxiliary variables on the boundary have to be introduced and trial and test functions of higher smoothness along the boundary are not required as a consequence the method leads to lower computational costs for discretizations with higher order elements and is easily integrated in high order finite element libraries a priori h convergence error estimates show that the method does not reduce the order of convergence compared to usual dirichlet neumann or robin boundary conditions if the polynomial degree on the boundary is increased simultaneously a series of numerical experiments illustrates the utility of the method and validates the theoretical convergence results

on the bohnenblust hille inequality and a variant of littlewoodʼs 4 3 inequality the search for sharp constants for inequalities of the type littlewood s 4 3 and bohnenblust hille has lately shown unexpected applications in many fields such as analytic number theory quantum information theory or in results on n dimensional bohr radii recent estimates obtained for the multilinear bohnenblust hille inequality for real scalars have been used as a crucial tool by a montanaro in order to solve problems in quantum xor games here among other results we obtain new upper bounds for the bohnenblust hille constants for complex scalars for bilinear forms we provide optimal constants of variants of littlewood s 4 3 inequality for real scalars when the exponent 4 3 is replaced by any r 4 3 we also prove that the optimal constants in real case are always strictly greater than those from the complex case

a summary of infinite element formulations for exterior helmholtz problems this work is devoted to a study and summary of different infinite element ie formulations for helmholtz problems in arbitrary exterior domains the theoretical setting for each of the different formulations is presented and related to the mathematical existence theory the influence of a bilinear or a sesquilinear formulation is discussed as well as possible extensions to other elements the implementation of the infinite element method iem incorporates the use of 2d and 3d hp finite elements and allows for hp adaptive refinements numerical results show the computational efficiency of the coupled finite infinite element methodology

the bohnenblust hille inequality for homogeneous polynomials is hypercontractive the bohnenblust hille inequality says that the 2m m 1 norm of the coefcients of an m homogeneous polynomial p on c n is bounded by kpk1 times a constant independent of n wherekk 1 denotes the supremum norm on the polydisc d n the main result of this paper is that this inequality is hypercontractive i e the constant can be taken to be c m for some c 1 combining this improved version of the bohnenblust hille inequality with other results we obtain the following the bohr radius for the polydisc d n behaves asymptotically as p logn n modulo a factor bounded away from 0 and innity

perfectly matched layers for elastic waves in cylindrical and spherical coordinates the perfectly matched layer pml for elastic waves in cylindrical and spherical coordinates is developed using an improved scheme of complex coordinates as is known for electromagnetic waves berenger s original pml scheme does not apply to cylindrical and spherical coordinates the straightforward extension of the complex coordinates for elastic waves to cylindrical and spherical coordinates requires extra unknowns for time domain solutions wasting computer memory and computation time the main idea of the improved scheme in this work is the use of integrated complex variables it is shown that for three dimensional cylindrical and spherical coordinates this improved pml scheme requires no more unknowns than in cartesian coordinates the number of unknowns can be further reduced through the use of symmetry in the partial differential equations the pml scheme allows an arbitrary inhomogeneity in the medium and is suitable for numerical solutions of wave equations by finite difference finite element

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for elastodynamics the scaled boundary finite element method alias the consistent infinitesimal finite element cell method is developed starting from the governing equations of linear elastodynamics only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one no fundamental solution is necessary and thus no singular integrals must be evaluated general anisotropic material is analysed without any increase in computational effort boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization this method is exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions for a bounded medium symmetric static stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption a stress singularity is represented very accurately as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization

special finite elements for use with high order boundary conditions abstract a hierarchy of special two dimensional finite elements is constructed for use in a layer near a boundary or an interface b the construction of the elements guarantees high order c n regularity along b and c 0 regularity elsewhere all the elements in the hierarchy are proved to be fully conforming the most important finite element in this class is the first order one which provides c 1 regularity on b and c 0 regularity elsewhere it is used here to improve the numerical solution of dirichlet problems by prescribing the tangential derivative of the solution on b in addition to the solution itself this element and higher order ones are also useful near artificial boundaries where high order local non reflecting boundary conditions are used and as transition elements near the interface between a domain governed by a c 0 theory and a domain governed by a c n theory

mixed finite element method and high order local artificial boundary conditions for exterior problems of elliptic equation the mixed finite element method is used to solve the exterior elliptic problem with high order local artificial boundary conditions new unknowns are introduced to reduce the order of the derivatives to two this leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary error estimates are obtained for some local artificial boundary conditions with prescribed order numerical examples are presented and the results demonstrate the effectiveness of this method

on the growth of the optimal constants of the multilinear bohnenblust hille inequality let kn n 1 be the optimal constants satisfying the multilinear real or complex bohnenblust hille inequality the exact values of the constants kn are still waiting to be discovered since eighty years ago with the publication of bohnenblust and hille paper in the annals of mathematics recently it was proved that kn n 1 has a subpolynomial growth moreover it is now known that if there is an l such that limn kn kn 1 l then l 0 in this note we show that if there is an l 0 such that limn k2nkn l then l 1 d with d e12 12 for real scalars and d e1 12 2 for complex scalars here is the famous euler mascheroni constant we show that this result generalizes the former

a nitsche method for wave propagation problems in time domain we propose a new nitsche type approach for the weak enforcement of dirichlet and neumann boundary conditions in the context of time domain wave propagation problems in mixed form a peculiar feature of the proposed method is that due to the hyperbolic structure of the problem considered two penalty parameters are introduced corresponding to dirichlet and neumann conditions respectively a stability and convergence estimate is also provided in the case of a discontinuous in time galerkin space time integrator the spatial discretization used is based on a stabilized method with equal order interpolation for all solution components in principle however the proposed methodology is not confined to stabilized methods we conclude with an extensive set of tests to validate the robustness and accuracy of the proposed approach

numerical comparison of high order absorbing boundary conditions and perfectly matched layers for a dispersive one dimensional medium abstract high order absorbing boundary conditions abc and perfectly matched layers pml are two powerful methods to numerically solve wave problems in unbounded domains the aim of the proposed study is to analyze and compare the performance of these methods in the one dimensional problem governed by the dispersive wave equation the pmls proposed in literature for time harmonic dynamics are applied to time dependent wave problems and linear quadratic and cubic polynomial stretching functions are considered the resulting pmls exhibit a double absorbing action i they reduce the amplitude of the incident waves and ii they delay the wave propagation then the abc proposed by givoli and neta and those proposed by hagstrom mar or and givoli are considered the former are a reformulation of the higdon high order non reflecting boundary conditions the latter improve the higdon conditions and extend them to take into account evanescent waves the accuracy of the pmls and abcs implemented in a finite element code is first investigated with respect to the frequency of the incident wave being it progressive or evanescent then the response to a wave train characterized by a broad frequency spectrum resulting from an impulsive force is studied a detailed analysis is performed to detect the influence of the parameters of both the abcs and the pmls on the absorption of waves the performances of pmls and abcs are compared and merits and drawbacks of the two methods are pointed out