global solutions for a semilinear two dimensional klein gordon equation with exponential type nonlinearity we prove the existence and uniqueness of global solutions for a cauchy problem associated to a semilinear klein gordon equation in two space dimensions our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method 2006 wiley periodicals inc

the existence and global exponential stability of periodic solution for a neutral coupled system on networks with delays in this paper we establish sufficient conditions for the existence and global exponential stability of periodic solution to a type of neutral coupled system on networks with delays the key to prove the existence of periodic solutions is using the combined method of graph theory coincidence degree theory and lyapunov functional method and the sufficient conditions are easy to be checked finally a numerical simulation is carried out to show the correctness of our main results

preconditioning and iterative solution of all at once systems for evolutionary partial differential equations standard krylov subspace solvers for self adjoint problems have rigorous convergence bounds based solely on eigenvalues however for non self adjoint problems eigenvalues do not determine behavior even for widely used iterative methods in this paper we discuss time dependent pde problems which are always non self adjoint we propose a block circulant preconditioner for the all at once evolutionary pde system which has block toeplitz structure through reordering of variables to obtain a symmetric system we are able to rigorously establish convergence bounds for minres which guarantee a number of iterations independent of the number of time steps for the all at once system if the spatial differential operators are simultaneously diagonalizable we are able to quickly apply the preconditioner through use of a sine transform and for those that are not we are able to use an algebraic multigrid process to provide a good approximation results are presented for solution to both the heat and convection

multiple solutions to a linearly coupled elliptic system with critical exponents abstract we are concerned with the linearly coupled elliptic system with critical exponents u a x u u 2 2 u u p 2 u v x r n v b x v v 2 2 v v q 2 v u x r n u v h 1 r n where n 3 2 p q 2 2 n n 2 and are nonnegative constants and a x b x are positive periodic functions the existence of a positive ground state and a higher energy solution of this system is proved using variational methods additionally the asymptotic behavior of these solutions as 0 is obtained

periodic solutions of a second order nonlinear system abstract we discuss the existence of periodic solutions of the system x t d dt grad f x t g t x t e t under various asymptotic behaviours of g

the alternating linear scheme for tensor optimization in the tensor train format recent achievements in the field of tensor product approximation provide promising new formats for the representation of tensors in form of tree tensor networks in contrast to the canonical r term representation candecomp parafac these new formats provide stable representations while the amount of required data is only slightly larger the tensor train tt format siam j sci comput 33 2011 pp 2295 2317 a simple special case of the hierarchical tucker format j fourier anal appl 5 2009 p 706 is a useful prototype for practical low rank tensor representation in this article we show how optimization tasks can be treated in the tt format by a generalization of the well known alternating least squares als algorithm and by a modified approach mals that enables dynamical rank adaptation a formulation of the component equations in terms of so called retraction operators helps to show that many structural properties of the original problems transfer to the micro iterations giving what is to our knowledge the first stable generic algorithm for the treatment of optimization tasks in the tensor format for the examples of linear equations and eigenvalue equations we derive concrete working equations for the micro iteration steps numerical examples confirm the theoretical results concerning the stability of the tt decomposition and of als and mals but also show that in some cases high tt ranks are required during the iterative approximation of low rank tensors showing some potential of improvement

maximization problem on trudinger moser inequality involving lebesgue norm abstract we study a maximization problem on the trudinger moser inequality and clarify the effect of lower order perturbation on the attainability of the best constant in particular we show the explicit forms of threshold nonlinearity involving lower order perturbation our results are based on the estimates of the dirichlet energy of maximizers

periodic solutions of linear and quasilinear neutral functional differential equations abstract in this paper we will use the fourier expansions to generalize hale s alternative results for the existence of periodic solutions of linear retarded functional differential equations to neutral equations using some a priori estimation and the leray schauder degree theory we will obtain some existence theorems of periodic solutions for a class of quasilinear neutral equations the results also generalize the works of hatvani and krisztin on retarded equations

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth abstract in this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in r 2 u v x u 1 x f u x f u x e h x in r 2 where v is a positive continuous potential 0 2 0 2 2 e is a small parameter and f s is the primitive function of f s suppose that the nonlinearity f s is of critical exponential growth in the sense of trudinger moser inequality we prove the existence and multiplicity of solutions by variational methods

adams inequality with the exact growth condition in ℝ4 adams inequality is an extension of the trudinger moser inequality to the case when the sobolev space considered has more than one derivative the goal of this paper is to give the optimal growth rate of the exponential type function in adams inequality when the problem is considered in the whole space 4 2014 wiley periodicals inc

bounds for best constants in subcritical sobolev embeddings abstract we establish upper and lower estimates for the embedding constants related to the classical sobolev embeddings h 0 1 l p where r n is a bounded domain or the whole r n and p 2 2 n n 2 n 3 in dimension n 2 we also derive the sharp asymptotic behavior of the embedding constant as p

existence and concentration of ground state solutions for a critical nonlocal schrodinger equation in r2 we study the following singularly perturbed nonlocal schrodinger equation e2 u v x u e 2 1 x f u f u inr2 where v x is a continuous real function on r2 f s is the primitive of f s 0 2 and e is a positive parameter assuming that the nonlinearity f s has critical exponential growth in the sense of trudinger moser we establish the existence and concentration of solutions by variational methods

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

attainability of the best sobolev constant in a ball the best constant in the sobolev inequality in the whole space is attained by the aubin talenti function however this does not happen in bounded domains because of the break down of the dilation invariance in this paper we investigate a new scale invariant form of the sobolev inequality in a ball and show that its best constant is attained by functions of the aubin talenti type generalization to the caffarelli kohn nirenberg inequality in a ball is also discussed

fast iterative solvers for buoyancy driven flow problems we outline a new class of robust and efficient methods for solving the navier stokes equations with a boussinesq model for buoyancy driven flow we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters

multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent abstract this paper deals with the following system linearly coupled by nonlinear elliptic equations u 1 u u 2 2 u v x v 2 v v 2 2 v u x u v 0 on here is a smooth bounded domain in r n n 3 1 2 1 are constants 1 is the first eigenvalue of h 0 1 2 2 n n 2 is the sobolev critical exponent and r is a coupling parameter by variational method we prove that this system has a positive ground state solution for some 0 via a perturbation argument we show that this system also admits a positive higher energy solution when is small moreover the asymptotic behaviors of the positive ground state and higher energy solutions as 0 are analyzed

bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two abstract we prove using variational methods the existence in dimension two of positive vector ground states solutions for the bose einstein type systems u 1 u 1 u e u 2 1 v e u v 1 in v 2 v 2 v e v 2 1 u e u v 1 in u v h 0 1 where is a bounded smooth domain 1 2 1 the first eigenvalue of h 0 1 1 2 0 and is either positive small or large or negative small the nonlinear interaction between two bose fluids is assumed to be of critical exponential type in the sense of j moser for small solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power like nonlinearities

new solutions for trudinger moser critical equations in r2 abstract let be a bounded smooth domain in r 2 we consider critical points of the trudinger moser type functional j u 1 2 u 2 2 e u 2 in h 0 1 namely solutions of the boundary value problem u u e u 2 0 with homogeneous dirichlet boundary conditions where 0 is a small parameter given k 1 we find conditions under which there exists a solution u which blows up at exactly k points in as 0 and j u 2 k we find that at least one such solution always exists if k 2 and is not simply connected if has d 1 holes in addition d 1 bubbling solutions with k 1 exist these results are existence counterparts of one by druet in o druet multibump analysis in dimension 2 quantification of blow up levels duke math j 132 2 2006 217 269 which classifies asymptotic bounded energy levels of blow up solutions for a class of nonlinearities of critical exponential growth including this one as a prototype case

revisiting an idea of brezis and nirenberg abstract let n 3 and be a c 1 bounded domain in r n with 0 suppose is c 2 at 0 and the mean curvature of at 0 is negative we prove the existence of positive solutions for the equation 0 1 u u n 2 n 2 u 2 s 1 x s 0 in u 0 on where 0 0 s 2 2 s 2 n s n 2 and n 4 for n 3 the existence result holds for 0 s 1 under the same assumption of the domain for p 2 s 1 we also prove the existence of a positive solution for the following equation 0 2 u u p u 2 s 1 x s 0 in u 0 on where 0 and 1 p n n 2

equivalent moser type inequalities in r2 and the zero mass case abstract we first investigate concentration and vanishing phenomena concerning moser type inequalities in the whole plane which involve complete and reduced sobolev norms in particular we show that the critical ruf inequality is equivalent to an improved version of the subcritical adachi tanaka inequality which we prove to be attained then we consider smooth compactly supported functions with respect to the dirichlet norm 2 and we prove an optimal lorentz zygmund type inequality with explicit extremals and from which can be derived classical inequalities in h 1 r 2 such as the adachi tanaka inequality and a version of ruf s inequality

estimates of the conformal scalar curvature equation via the method of moving planes in this paper we derive a local estimate of a positive singular solution u near its singular set z of the conformal equation where k x is a positive continuous function z is a compact subset of and g satisfies that is nonincreasing for t 0 assuming that the order of flatness at critical points of k on z is no less than we prove that through the application of the method of moving planes the inequality holds for any solution of 0 1 with cap z 0 by the same method we also derive a harnack type inequality for smooth positive solutions let u satisfy assume that the order of flatness at critical points of k is no less than n 2 then the inequality holds for r 1 we also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality 0 4 1997 john wiley sons inc

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

sharp exponential integrability for critical riesz potentials and fractional laplacians on rn abstract we derive sharp adams inequalities for the riesz and more general riesz like potentials on the whole of r n as a consequence we obtain sharp moser trudinger inequalities for the critical sobolev spaces w n r n 0 n these inequalities involve fractional laplacians higher order gradients general homogeneous elliptic operators with constant coefficients and general trace type borel measures

two periodic solutions of second order neutral functional differential equations in this paper we consider a type of second order neutral functional differential equations we obtain some existence results of multiplicity and nonexistence of positive periodic solutions our approach is based on a fixed point theorem in cones

on the existence of periodic solutions to rayleigh differential equation of neutral type in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t c x t f x t g t x t t e t in the critical case c 1 by analyzing some properties of the linear difference operator a a x t x t c x t and using mawhin s continuation theorem some new results are obtained

adaptive time stepping for incompressible flow part ii navier stokes equations we outline a new class of robust and efficient methods for solving the navier stokes equations we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the potential of our approach

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

on a weighted trudinger moser inequality in rn abstract we establish the trudinger moser inequality on weighted sobolev spaces in the whole space and for a class of quasilinear elliptic operators in radial form of the type l u r r u r u r where 0 and 0 are constants satisfying some existence conditions it is worth emphasizing that these operators generalize the p laplacian and k hessian operators in the radial case our results involve fractional dimensions a new weighted polya szego principle and a boundness value for the optimal constant in a gagliardo nirenberg type inequality

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

periodic solutions for a kind of second order differential equation with multiple deviating arguments by means of continuation theorem of coincidence degree theory some new results on the existence and nonexistence of periodic solutions for a kind of second order functional differential equation with multiple deviating arguments are obtained

periodic solutions for some nonlinear differential equations of neutral type where f ir x ir ir is a caratheodory function f x is 2n periodic for every x ci e r and r e io 2n this problem has been studied by sadovskii in 151 by hale and mawhin in 6 in the case ial 1 to the previous one our interest is particularly motivated by the fact that represents a simple and meaningful model for a general phenomenon which often arises in the study of functional differential equations in fact as is well known in some cases a delayed differential equation can be seen as a perturbation of an ordinary differential equation in particular this occurs when the delayed term in the equation can be considered smaller than the ordinary one or dominated by it when such domination is available many good properties can be proved and several results such as those of 6 and 16 take advantage of this fact on the contrary when no domination is present the situation is much more complicated and in general very little is known in problem the delayed term can be considered dominated by the ordinary one whenever ial 1 as a matter of fact in this case one can prove the invertibility of the differential operator l x x a r ux the compactness of its inverse and even some a priori estimates for the solutions consequently one can reduce to a fixed point problem and solve it by means of some topological degree argument however when 11 1 so that the weight of the two terms x t and x t t is the same the above properties fail and moreover for some delays the kernel of the differential operator lx x a r is infinite dimensional therefore one must look for different methods problem is thus a simple but nevertheless significant example of the considerations illustrated by hale in 5 about the difficulty of undertaking a unified approach in the study of functional differential equations at the same time it provides an analogy between neutral equations and the wave equation in the line suggested by hale in 5

alternating minimal energy methods for linear systems in higher dimensions we propose algorithms for the solution of high dimensional symmetrical positive definite spd linear systems with the matrix and the right hand side given and the solution sought in a low rank format similarly to density matrix renormalization group dmrg algorithms our methods optimize the components of the tensor product format subsequently to improve the convergence we expand the search space by an inexact gradient direction we prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm the complexity of the presented algorithms is linear in the mode size and dimension and the demonstrated convergence is comparable to or even better than the one of the dmrg algorithm in the numerical experiment we show that the proposed methods are also efficient for non spd systems for example those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale

periodic solutions for generalized high order neutral differential equation in the critical case abstract by applying mawhin s continuation theory and some new inequalities we obtain sufficient conditions for the existence of periodic solutions for a generalized high order neutral differential equation in the critical case moreover an example is given to illustrate the results

ground states for a system of schrodinger equations with critical exponent abstract we study the following system of nonlinear schrodinger equations u u u p 1 u v x r n v v v 2 2 v u x r n where n 3 2 2 n n 2 1 p 2 1 and are positive parameters satisfying 0 we show that there is some critical value 0 0 1 such that this system has a positive ground state solution if 0 0 in the case 0 there exists 0 such that this system has no ground state solutions if while this system has a positive ground state solution if in particular if p 2 1 the system has no nontrivial solutions some further properties of the ground state solutions are also studied this seems to be the first result for such a critical schrodinger system

lorentz sobolev spaces and systems of schrodinger equations in rn abstract we study the connection between the improvement of limiting sobolev s embeddings within the context of lorentz spaces and the variational approach to systems of nonlinear schrodinger equations we show that lorentz sobolev spaces appear as a natural function space domain for the related energy functional moreover in this framework the nonlinearity may exhibit a supercritical growth with respect to the maximal growth prescribed by the poho aev trudinger moser inequality and still preserving a variational structure

periodic solutions to a second order nonlinear neutral functional differential equation in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t cx t f x t x t g t x t t e t in the critical case c 1 by employing mawhin s continuation theorem and some analysis techniques sufficient conditions are given for the existence of periodic solutions

positive ground state solutions for a nonlinearly coupled schrodinger system with critical exponents in r4 abstract consider the following nonlinear schrodinger equations u 1 u 1 u 3 3 u 2 v u v 2 v 3 x v 2 v 2 v 3 3 u v 2 u 2 v u 3 x u 0 v 0 in u v 0 on where r 4 is a smooth bounded domain 1 2 0 1 2 0 0 and is the first eigenvalue of with the dirichlet boundary condition note that all the cubic nonlinearities including the coupling terms are of critical growth the existence of positive ground state solutions for the system is established

stochastic galerkin methods for the steady state navier stokes equations we study the steady state navier stokes equations in the context of stochastic finite element discretizations specifically we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion for the resulting stochastic problem we formulate the model and linearization schemes using picard and newton iterations in the framework of the stochastic galerkin method and we explore properties of the resulting stochastic solutions we also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic galerkin nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems

analysis of a new space time parallel multigrid algorithm for parabolic problems we present and analyze a new space time parallel multigrid method for parabolic equations the method is based on arbitrarily high order discontinuous galerkin discretizations in time and a finite element discretization in space the key ingredient of the new algorithm is a block jacobi smoother we present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate we then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties

periodic solutions to a kind of neutral functional differential equation in the critical case in this paper we consider a kind of neutral functional differential equation as follows x t cx t gt xt t e t in the critical case c 1 by employing the topological degree theory and some analysis techniques we obtain some new results on the existence of periodic solutions

on the existence of periodic solutions for neutral functional differential equation abstract by means of mawhin s continuation theorem existence criteria are established for the periodic solutions of neutral functional differential equation

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

symmetry results for semilinear elliptic systems in the whole space in this paper we obtain symmetry results of gidas ni nirenberg type for cooperative nonlinear elliptic systems in the whole space

a low rank solver for the stochastic unsteady navier stokes problem abstract we study a low rank iterative solver for the unsteady navier stokes equations for incompressible flows with a stochastic viscosity the equations are discretized using the stochastic galerkin method and we consider an all at once formulation where the algebraic systems at all the time steps are collected and solved simultaneously the problem is linearized with picard s method to efficiently solve the linear systems at each step we use low rank tensor representations within the krylov subspace method which leads to significant reductions in storage requirements and computational costs combined with effective mean based preconditioners and the idea of inexact solve we show that only a small number of linear iterations are needed at each picard step the proposed algorithm is tested with a model of flow in a two dimensional symmetric step domain with different settings to demonstrate the computational efficiency

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

preconditioning steady state navier stokes equations with random data we consider the numerical solution of the steady state navier stokes equations with uncertain data specifically we treat the case of uncertain viscosity which results in a flow with an uncertain reynolds number after linearization we apply a stochastic galerkin finite element method combining standard inf sup stable taylor hood approximation on the spatial domain on highly stretched grids with orthogonal polynomials in the stochastic parameter this yields a sequence of nonsymmetric saddle point problems with kronecker product structure the novel contribution of this study lies in the construction of efficient block triangular preconditioners for these discrete systems for use with gmres crucially the preconditioners are robust with respect to the discretization and statistical parameters and we exploit existing deterministic solvers based on the so called pressure convection diffusion and least squares commutator approximations

sharp adams type inequalities in ℝⁿ adams inequality for bounded domains r4 states that the supremum of e 32 u dx over all functions u w 2 2 0 with u 2 1 is bounded by a constant depending on only this bound becomes infinite for unbounded domains and in particular for r4 we prove that if u 2 is replaced by a suitable norm namely u u u 2 then the supremum of e 32 u 1 dx over all functions u w 2 2 0 with u 1 is bounded by a constant independent of the domain furthermore we generalize this result to any w m n m 0 with rn and m an even integer less than n

existence and nonexistence of maximizers for variational problems associated with trudinger moser type inequalities in mathbb r n we discuss the existence of a maximizer for a maximizing problem associated with the trudinger moser type inequality in mathbb r n n geq 2 different from the bounded domain case we obtain both of the existence and the nonexistence results the proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences

standing waves for nonlinear schrodinger equations with a general nonlinearity for elliptic equations e2 u v x u f u 0 x rn n 3 we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of v as e 0 under conditions on f which we believe to be almost optimal

existence and non existence of maximizers for the moser trudinger type inequalities under inhomogeneous constraints in this paper we study the existence and non existence of maximizers for the moser trudinger type inequalities in mathbb r n of the form begin aligned d n alpha a b sup u in w 1 n mathbb r n vert nabla u vert l n mathbb r n a vert u vert l n mathbb r n b 1 int mathbb r n phi n left alpha u n right dx end aligned here n ge 2 n frac n n 1 a b 0 alpha in 0 alpha n and phi n t e t sum j 0 n 2 frac t j j where alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit ball in mathbb r n we show the existence of the threshold alpha alpha a b n in 0 alpha n such that d n alpha a b is not attained if alpha in 0 alpha and is attained if alpha in alpha alpha n we also provide the conditions on a b in order that the inequality alpha alpha n holds