provably good global routing of integrated circuits this paper investigates the global routing problem for integrated circuits we introduce a formulation on the basis of integer programming which minimizes the routing area among a limited set of steiner trees for each net indeed the involved cost function depends on the channel density of the routing which has a direct influence on the routing area our methods for solving the global routing problem employ local search heuristics sequential routing genetic routing and randomized procedures our methods for computing lower bounds are based on linear and lagrange relaxation an analysis on the tightness of the bounds indicates that the difference between the cost of the optimal integer solutions and the cost of the optimal fractional solutions is only a small number of tracks in practice moreover the analysis leads to the concept of linear preprocessing by which we exclude a large number of high cost solutions we introduce several versions of preprocessing one of which preserves the opportunity of obtaining a globally optimal solution in general all of them do so in practice linear preprocessing enables us to solve problem instances with several thousand nets provably optimal or at least provably close to optimal all methods have been implemented in the software package eridanus we present computational results the global routing problem assumes the placement of the chip components to be fixed an extension of the problem which we call global layout of integrated circuits allows the placement to be variable and searches for a placement that minimizes the routing area among a limited set of alternatives we show that the results concerning the global routing problem can be easily extended to global layout

on the physical significance of higher order kinematic and static variables in a three dimensional shell formulation abstract in recent years considerable attention has been given to the development of higher order plate and shell models these models are able to approximately represent three dimensional effects while pertaining the efficiency of a two dimensional formulation due to pre integration of the structural stiffness matrix across the thickness especially the possibility to use unmodified complete three dimensional material laws within shell analysis has been a major motivation for the development of such models while the theoretical and numerical formulation of so called 7 parameter shell models including a thickness stretch of the shell has been discussed in numerous papers no thorough investigation of the physical significance of the additional kinematic and static variables coming along with the extension into three dimensions is known to the authors however realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures e g concerning loading and kinematic boundary conditions and a correct interpretation of the results in the present paper the significance of kinematic and static variables appearing in a 7 parameter model proposed by buchter and ramm 1992a are discussed it is shown how these quantities refine the model behavior and how they can be related to the classical variables such as curvatures and stress resultants furthermore the special role of the material law within such a formulation is addressed it is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness to ensure correct results in this context it is found that the considered 7 parameter model can be regarded as optimal with respect to the number of degrees of freedom involved

adaptive computational meso macro scale modeling of elastic composites we consider combined material model and fe mesh adaptivity in the context of computational meso macro scale modeling cmm whereby the coupling between different geometrical scales is taken into account the macro scale model is then replaced by the homogenized response of a representative volume element rve in practice the rve is analyzed as a small scale finite element problem which is solved for each gauss point in the domain the high cost of this strategy clearly motivates the use of adaptive techniques based on a posteriori error estimates for the combined model and discretization error whereby the solution of an appropriate dual problem is utilized the suggested adaptive procedure is applied to a particle matrix composite with hyperelastic constituents

existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

applying the hp d version of the fem to locally enhance dimensionally reduced models abstract a modification of the hp d method is presented which allows to locally enhance a dimensionally reduced finite element approximation with an ansatz based on a more accurate i e dimensionally higher model the procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models like beams plates and solids thereby the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model two examples are given to demonstrate the basic properties of the proposed method in the first example the solution of a one dimensional beam model is improved locally by a two dimensional finite element computation in the second example the local enhancement of a reissner mindlin plate by a fully three dimensional continuum approach is considered yielding a significant reduction of the model error

existence of ground states for a kirchhoff type problem without 4 superlinear condition abstract we consider the existence of ground state solutions for the kirchhoff type problem a b r n u 2 d x u v x u u p 2 u x r n u h 1 r n where a b 0 n 1 2 3 and 2 p 2 here we are interested in the case that 2 p 4 since the existence of ground state for 4 p 2 is easily obtained by a standard variational argument our method is based on a poho z aev type identity

local error estimation for dimensionally reduced models of elliptic boundary value problems in this paper we present an approach to the local estimation of errors in dimensionally reduced models of elliptic boundary value problems posed on thin domains in this approach errors are measured in quantities of interest whose computation is typically the goal of the analysis explicit bounds and estimates are derived for errors in quantities characterized by continuous linear functionals a representative numerical example showing the accuracy of the bounds and estimates is also presented

application of an automatic hp adaptive finite element method for thin walled structures the paper investigates the possibility of solving beam and thin walled structures with an automatic hp adaptive finite element method for three dimensional linear elasticity this technique generates sequences of meshes with nonuniform distribution of element sizes h and their spectral orders p to minimize the interpolation error for the hypothetical exact solution with constant number of degrees of freedom the intent of using hp method for beam and thin walled structures is that it should automatically create fe meshes with adequate distribution of the approximation orders to prevent the numerical locking phenomenon and to provide accurate solutions the method also reduces the human work spent on building the numerical model to generation of the description of the geometry we discuss energy based and goal oriented versions of hp adaptivity the paper presents a broad spectrum of examples of engineering applications

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

on a variational inequality for a degenerate quasilinear hyperbolic equation we study a unilateral problem for the nonhomogeneous degenerated kirchhoff equation with a blowing up term making use of the penalty method and galerkin s approximations we establish global existence and uniqueness theorems

an estimate of the interval of existence of solutions of the nonlinear kirchhoff equation abstract solutions to the initial boundary value problem for the nonlinear kirchhoff equation are considered conditions on the initial data and nonlinear term are given so that the solutions to the problem under consideration do not exist for all t 0 an estimate of the t interval of existence of the solutions is obtained

asymptotic analysis of linearly elastic shells iii justification of koiter s shell equations we consider as in part i a family of linearly elastic shells of thickness 2 all having the same middle surfaces r3 where r2 is a bounded and connected open set with a lipschitz continuous boundary and l3 r3 the shells are clamped on a portion of their lateral face whose middle line is 0 where 0 is any portion of withlength 0 0 we make an essential geometrical assumption on the middle surfaces and on the set 0 which states that the space of inextensional displacements begin gathered v f omega eta eta i in h 1 omega times h 1 omega times h 2 omega hfill eta i partial v eta 3 0 on gamma 0 gamma alpha beta eta 0 in omega hfill end gathered where gamma alpha beta are the components of the linearized change is metric tensor ofs contains non zero functions this assumption is satisfied in particular ifs is a portion of cylinder and 0 is contained in a generatrix ofs

the existence of least energy nodal solutions for some class of kirchhoff equations and choquard equations in r n abstract in this paper we study the existence of least energy nodal solutions for a class of kirchhoff type problems in r n since the kirchhoff equation is a nonlocal one the variational setting to look for sign changing solutions is different from the local cases by using constrained minimization on the sign changing nehari manifold we prove the kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy as a co product of our approaches we obtain the existence of least energy sign changing solution for choquard equations in r n and show that the sign changing solution has an energy strictly larger than the least energy and less than twice the least energy

a priori error estimations of hp finite element approximations for hierarchical models of plate and shell like structures a priori error estimates of hp finite element approximations for hierarchical models of the thin elastic plate and shell like structures are derived the error in the approximations of a hierarchical model consists of the modelling error due to assumptions on displacement or stress fields and the finite element approximation error of discretizations of such a model also non uniformity with respect to the thickness owing to numerical locking is included in the present error estimates numerical results are given supporting the theoretical results

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

pq adaptive solid finite elements for three dimensional plates and shells this article presents a pq adaptive finite element procedure for three dimensional computation of thin walled structures like plates and shells the approach is based on a hexahedral element with high order shape functions the polynomial degree of the shape functions in the three spatial directions of the element is adjusted by means of an adaptive procedure to this end a hierarchical error indicator is applied to automatically control the polynomial degree q in thickness and p in the in plane direction of plate and shell like structures the efficiency and reliability of the proposed adaptive method is demonstrated using two numerical examples

low regularity global solutions for nonlinear evolution equations of kirchhoff type we study the global solvability of the cauchy dirichlet problem for two second order in time nonlinear integro differential equations 1 the extensible beam plate equation utt 2u m u 2dx u 0 x t r 2 a special case of the kirchhoff equation utt a b u 2dx 2 u 0 x t r by exploiting the i method of j colliander m keel g staffilani h takaoka t tao we prove that both equations admit global in time infinite energy solutions in case 1 the energy is the mechanical energy in case 2 it is a second order invariant introduced by s i pokhozhaev for the extensible beam equation 1 we also consider the effect of linear dissipation on such low regularity solutions and we prove their exponential decay as t

the maximum concurrent flow problem the maximum concurrent flow problem mcfp is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently the ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow the mcfp objective is to maximize the throughput subject to the capacity constraints we develop a fully polynomial time approximation scheme for the mcfp for the case of arbitrary demands and uniform capacity computational results are presented it is shown that the problem of associating costs distances to the edges so as to maximize the minimum cost of routing the concurrent flow is the dual of the mcfp a path cut type duality theorem to expose the combinatorial structure of the mcfp is also derived our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm applications include packet switched networks 1 4 8 and cluster analysis 16

signed and sign changing solutions for a kirchhoff type equation in bounded domains abstract the main concern of this article is a kirchhoff type equation of the form m u 2 u f u where is a bounded smooth domain in r n with n 3 and is a positive parameter under certain assumptions on m and f the existence results of signed and sign changing solutions are established for large and when converges to infinity the asymptotic behavior of these solutions is also studied the proofs are based on a careful study of the ground state and least energy nodal solutions of an auxiliary problem which is constructed by making a refined truncation on m furthermore we get the ground state and least energy nodal solutions and prove the energy doubling property for all 0 under more restricted assumptions on m and f

existence and multiplicity of nontrivial solutions for a class of schrodinger kirchhoff type equations abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u f x u in r n u x 0 as x is studied when n 2 3 4 and v x 1 the existence theorem of nontrivial weak solutions for problem p is obtained when n 1 2 3 and v x is more general two existence theorems of nontrivial weak solutions and a sequence of high energy weak solutions for problem p are obtained

the existence of a ground state solution for a class of kirchhoff type equations in ℝn in this paper we study the following kirchhoff type equation where a b are positive constants and n 1 2 3 under appropriate assumptions on v k and g we obtain a ground state solution by using the approach developed by szulkin and weth in 2010

the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations abstract this paper deals with the number of solutions for a class of nonlocal nonhomogeneous gradient operator equations of the form a i u i u f where i c 1 x r x is a reflexive banach space i 0 0 i is even and strictly convex i u u as u i x x is a bounded homeomorphism but is not necessarily homogeneous a 0 r is continuous f x 0 some properties and examples of such a functional i are given some results on the number of solutions of the nonlocal equation are obtained

global bifurcation and positive solution for a class of fully nonlinear problems in this paper we study global bifurcation phenomena for the following kirchhoff type problem m u x 2 d x u f x u in u 0 on where m is a continuous function under some natural hypotheses we show that 1 a m 0 0 is a bifurcation point and there is a global continuum c emanating from 1 a m 0 0 where 1 a denotes the first eigenvalue of the above problem with f x s a x s as an application of the above result we study the existence of positive solution for this problem with asymptotically linear nonlinearity

packing steiner trees a cutting plane algorithm and computational results in this paper we describe a cutting plane algorithm for the steiner tree packing problem we use our algorithm to solve some switchbox routing problems of vlsi design and report on our computational experience this includes a brief discussion of separation algorithms a new lp based primal heuristic and implementation details the paper is based on the polyhedral theory for the steiner tree packing polyhedron developed in our companion paper this issue and meant to turn this theory into an algorithmic tool for the solution of practical problems

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

hierarchic models for laminated composites the principles governing the formulation of hierarchic models for laminated composites are discussed the essential features of the hierarchic models described herein are a the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness and b the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity examples are presented

on a model connected with the kirchhoff equation abstract we prove a global existence result in suitable sobolev spaces for the solution of the nonlinear cauchy problem tt u t x 1 r n dy k x y y u t y 2 x u t x 0 u 0 x u 0 x t u 0 x u 1 x 0 where n 3 is the space dimension the initial data are c with compact support in rn and k z is a positive smooth function rapidly decreasing as z our approach is based on the energy estimates combined with the classical von wahl inequalities

the lax mizohata theorem for nonlinear gauge invariant equations we prove that ifthe cauchy problem for the nonlinear partial differential equation is well posed in a neighbourhood of a given solution then all the characteristics of the equation are real valued for the linear equations this is a statement of the lax mizohata theorem equations considered in the paper include gauge invariant localequations and systems and non local kirchhoff type equations

existence and asymptotic behavior of nodal solutions for the kirchhoff type problems in r3 abstract in this paper we study the existence and asymptotic behavior of nodal solutions to the following kirchhoff problem a b r 3 u 2 d x u v x u f x u in r 3 u h 1 r 3 where v x is a smooth function a b are positive constants because the so called nonlocal term r 3 u 2 d x u is involved in the equation the variational functional of the equation has totally different properties from the case of b 0 under suitable construction conditions we prove that for any positive integer k the problem has a sign changing solution u k b which changes signs exactly k times moreover the energy of u k b is strictly increasing in k and for any sequence b n 0 n there is a subsequence b n s such that u k b n s converges in h 1 r 3 to w k as s where w k also changes signs exactly k times and solves the following equation a u v x u f x u in r 3 u h 1 r 3

adaptive hpq finite element methods of hierarchical models for plate and shell like structures abstract a four step adaptive strategy for hpq finite element approximations of hierarchical models of the thin plate and shell like structures is derived based on a priori and a posteriori error estimations this strategy controls three parameters the model level q the mesh size h and the approximation order p so as to produce quasi optimal hierarchical models and finite element meshes numerical results are given supporting the theoretical results

a posteriori modeling error estimation for hierarchic plate models hierarchic plate models are dimensional reductions obtained by semidiscretization of the three dimensional plate problem in the transverse direction and energy projection we derive computable a posteriori estimators for the modeling error thus incurred under the assumption that the resulting two dimensional plate models are solved exactly the estimators are valid for homogeneous monoclinic materials for plates with unsmooth midsurfaces and for a wide class of variational edge conditions computable a posteriori bounds on the effectivity indices are also derived and sufficient conditions for the asymptotic i e as the plate thickness tends to zero and the spectral i e as the order of the plate model tends to infinity exactness of the estimators are given numerical examples are presented

hierarchical models of thin elastic structures overview and recent advances in error estimation and adaptivity abstract in the numerical approximation of hierarchical models of thin bodies there are two separate contributions to the total error a modeling error is the error due to dimensional reduction and b discretization error is the error due to the numerical approximation traditionally estimation of these errors has been performed in terms of abstract quantities like the global energy norm which does not give a clear representation of the error in the local features of the solution recently a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst following a brief overview of the literature we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning independent estimates are obtained for the modeling and discretization errors in quantities of interest these local estimates are used to develop goal oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy

an approximate algorithm for a kirchhoff wave equation the initial boundary value problem for a hyperbolic type quasilinear equation w tt varphi int 0 pi w x 2dx w xx is considered its solution is derived by means of a numerical algorithm consisting of the projection method and the difference method for approximation with respect to spatial and time variables while the resulting discrete system is solved by an iteration process the accuracy of the proposed algorithm is estimated

on the mathematical foundation of the 1 1 2 platemodel abstract the plate problem of three dimensional 3 d linearized elastostatics is considered inthe framework of the hierarchical modelling with the help of the energy projection method it isshown that the 1 1 2 bending model is the simplest asymptotically correct model in thehierarchical family i e the distance between the solution of the 3 d problem and themodel solution in the energy norm tends to zero for vanishing plate thickness h whileusing unmodified 3 d material laws the formulation of the 1 1 2 bending model and the mechanical significance of itsingredients are discussed we present error estimates for the deviation of the 1 1 2 solutionfrom the kirchhoff solution as well as from the 3 d solution by using the 1 1 2 energy normand the 3 d energy norm respectively the analysis leads to h as order of convergence the results are illustrated by a numerical example

a numerical algorithm for the nonlinear kirchhoff string equation the initial boundary value problem is considered for the dynamic string equation its solution is found by means of an algorithm the constituent parts of which are the galerkin method the modified crank nicolson difference scheme used to perform approximation with respect to spatial and time variables and also a picard type iteration process for solving the system of nonlinear equations obtained by discretization errors of the three parts of the algorithm are estimated and as a result its total error estimate is obtained

on a p kirchhoff equation via fountain theorem and dual fountain theorem abstract in this paper we show the existence of infinite solutions to the kirchhoff type quasilinear elliptic equation m u p x u p d x p 1 p u x u p 2 u f x u in a smooth bounded domain r n with nonlinear boundary condition u p 2 u u p 2 on the method we used here is based on fountain theorem and dual fountain theorem

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

multiple nontrivial solutions to a pp kirchhoff equation abstract in this paper morse theory is used to show the existence of nontrivial solutions to the nonlocal p kirchhoff type quasilinear elliptic equation m u p d x p 1 p u f x u in a smooth bounded domain r n with dirichlet boundary condition

existence and nonexistence of positive solutions for nonlocal problems with inhomogeneous strong allee effect in this paper we deal with the nonlocal elliptic problem with inhomogeneous strong allee email protected w1p u email protected email protected x u email protected u 0 email protected email protected where the nonlocal coefficient email protected w1p u pdx is a continuous function of w12 u 2dx by means of variational approach we prove that the problem has at least two positive solutions for large l under suitable hypotheses about nonlinearity we also prove some nonexistence results

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

hierarchical models of elastic shells in curvilinear coordinates in the present paper static and dynamical problems for linearly elastic shells in curvilinear coordinates are considered hierarchies of two dimensional models for corresponding boundary and initial boundary value problems are constructed within the variational settings the existence and uniqueness of solutions of the reduced problems are investigated in suitable spaces under the conditions of solvability of the original static or dynamical problem convergence of the sequence of vector functions of three variables restored from the solutions of the constructed two dimensional problems to the solution of the three dimensional problem is proved and approximation error is estimated

multiplicity results for a p laplacian discrete problems of kirchhoff type this paper is concerned with the existence and multiplicity solutions for a discrete boundary value problem of p kirchhoff type our technical approach is based on variational methods

packing steiner trees polyhedral investigations letg v e be a graph andt v be a node set we call an edge sets a steiner tree fort ifs connects all pairs of nodes int in this paper we address the following problem which we call the weighted steiner tree packing problem given a graphg v e with edge weightswe edge capacitiesce e e and node sett1 tn find edge setss1 sn such that eachsk is a steiner tree fortk at mostce of these edge sets use edgee for eache e and the sum of the weights of the edge sets is minimal our motivation for studying this problem arises from a routing problem in vlsi design where given sets of points have to be connected by wires we consider the steiner tree packing problem from a polyhedral point of view and define an associated polyhedron called the steiner tree packing polyhedron the goal of this paper is to partially describe this polyhedron by means of inequalities it turns out that under mild assumptions each inequality that defines a facet for the single steiner tree polyhedron can be lifted to a facet defining inequality for the steiner tree packing polyhedron the main emphasis of this paper lies on the presentation of so called joint inequalities that are valid and facet defining for this polyhedron inequalities of this kind involve at least two steiner trees the classes of inequalities we have found form the basis of a branch cut algorithm this algorithm is described in our companion paper in this issue

kirchhoff type system with linear weak damping and logarithmic nonlinearities abstract for the nonlinear kirchhoff type wave system with logarithmic nonlinearities and weak dissipation the global well posedness of initial boundary value problem is analyzed focusing on the interplay between kirchhoff terms and logarithmic sources we investigate the kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest by employing potential well method and concavity method we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy which is used to classify initial data for global existence and finite time blow up finally via careful analysis involving the unstable invariant set under supercritical initial energy we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions

sign changing solutions for a class of kirchhoff type problem in bounded domains abstract we are interested in the existence of least energy sign changing solutions for a class of kirchhoff type problem in bounded domains because the so called nonlocal term b u 2 d x u is involving in the equation the variational functional of the equation has totally different properties from the case of b 0 combining constraint variational method and quantitative deformation lemma we prove that the problem possesses one least energy sign changing solution u b moreover we show that the energy of u b is strictly larger than the ground state energy finally we regard b as a parameter and give a convergence property of u b as b 0

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 1 hierarchical modeling and approximations this is a paper out of a series of three papers devoted to model and hpq adaptive finite element methods assigned for modeling and analysis of complex structures in this paper we focus our attention on hierarchical models and hierarchical approximations while the issues of error estimation and adaptivity control will be presented in the next papers of the series in all the papers we consider the elastic structures of complex mechanical description we apply 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within our finite element formulation in the proposed approach the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q can vary locally i e they can be different in each finite element the a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors the error controlled adaptive procedures are derived from a four step strategy with possible iterations on h and p steps

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

a perspective on adaptive modeling and meshing am m the purpose of this study is to develop an integrated approach to adaptive modeling in conjunction with adaptive meshing this involves consideration of the modeling error from various sources and the discretization error in the numerical scheme adapting the model to control error due to geometry boundary data constitutive relations and model parameters can be integrated with adaptive mesh control to improve numerical reliability other factors such as stabilization by artificial dissipation and regularization are considered a class of hybrid algorithms are developed to combine these approaches

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

3d based hierarchical models and hpq approximations for adaptive finite element method of laplace problems as exemplified by linear dielectricity abstract this paper is devoted to model adaptation and h p q adaptive finite element methods for modeling and analysis of the problems for which the strong formulation corresponds to laplace equation the chosen example of this equation concerns dielectric structures or media of electrostatics the paper addresses hierarchical theories also called hierarchical models and hierarchical approximations in the assessment of the models and approximations our own and existing a priori error estimation results are applied the used assessment procedure can be employed to any other applications of laplace equation in applied sciences the proposed theories understood as mathematical formulations and their numerical approximations are applied to the physical model of linear dielectricity in structures with complex electric description and complex geometry we take advantage of the 3d and 3d based theories hierarchical modeling and hierarchical approximations within h p q finite element formulation in our research the applied theory and discretization parameters i e the element size h the longitudinal approximation order p and the transverse order q differ in each finite element

new existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems abstract in the present paper we study the existence and multiplicity of nontrivial solutions for nonlocal elliptic kirchhoff type problems a b u 2 d x u f x u in u 0 on where a b 0 are real numbers some new existence and multiplicity results of nontrivial solutions are obtained by means of the mountain pass theorem and a cut off technique one known result is improved

a posteriori modeling error estimates for the assumption of perfect incompressibility in the navier stokes equation we derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible navier stokes equation real fluids are never perfectly incompressible but always feature at least some low amount of compressibility thus their behavior is described by the compressible navier stokes equation the pressure being a steep function of the density we rigorously estimate the difference between an approximate solution to the incompressible navier stokes equation and any weak solution to the compressible navier stokes equation in the sense of lions without assuming any additional regularity of solutions heuristics and numerical results suggest that our error estimates are of optimal order in the case of well behaved flows and divergence free approximations of the velocity field thus we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations

anisotropic mixed finite elements for elasticity summary in this paper we present a family of mixed finite elements which are suitable for the discretization of slim domains the displacement space is chosen as nedelec s space of tangential continuous elements whereas the stress is approximated by normal normal continuous symmetric tensor valued finite elements we show stability of the system on a slim domain discretized by a tensor product mesh where the constant of stability does not depend on the aspect ratio of the discretization we give interpolation operators for the finite element spaces and thereby obtain optimal order a priori error estimates for the approximate solution all estimates are independent of the aspect ratio of the finite elements copyright 2011 john wiley sons ltd

a posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains the boundary value problem of heat conduction in a three dimensional laminated plate is approximated by a hierarchy of two dimensional models computable a posteriors indicators and estimators of the modeling error in various norms are derived and their local spectral and asymptotic exactness is proved sharp estimates for their effectivity indices are also obtained

the critical problem of kirchhoff type elliptic equations in dimension four abstract we study the following kirchhoff type elliptic problem p a b u 2 d x u u q u 3 u 0 in u 0 on where r 4 is a bounded domain with smooth boundary moreover we assume a 0 b 0 and 1 q 3 in this paper we prove the existence of solutions of p our tools are the variational method and the concentration compactness argument for ps sequences

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case