on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

existence and multiplicity of solutions for p x curl systems arising in electromagnetism abstract in this paper we study the existence and multiplicity of solutions to a class of p x curl systems arising in electromagnetism the results obtained in this paper extend several contributions concerning the p curl operator and we focus on new existence results which are due to the presence of variable exponent to our best knowledge our results are new even in the semilinear case

double phase problems with variable growth abstract we consider a class of double phase variational integrals driven by nonhomogeneous potentials we study the associated euler equation and we highlight the existence of two different rayleigh quotients one of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues the notion of eigenvalue is understood in the sense of pairs of nonlinear operators as introduced by fucik necas soucek and soucek the analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p x laplace operator the generalized mean curvature operator or the capillarity differential operator with variable exponent the results contained in this paper complement the pioneering contributions of marcellini mingione et al in the field of variational integrals with unbalanced growth

the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

multiple solutions for a class of p x laplacian equations in involving the critical exponent in this paper we first establish a principle of concentration compactness in then based on this concentration compactness principle we study the existence of solutions for a class of p x lapl

on the eigenvalues of second order pseudospectral differentiation operators abstract recent results on the behavior of eigenvalues for the second order legendre spectral galerkin differentiation operator are extended to the eigenvalues of the corresponding legendre pseudospectral collocation operator numerical comparisons show that similar results also hold in the case of spectral chebyshev methods

comparison and sub supersolution principles for the fractional p x laplacian abstract in this paper we study the new fractional sobolev space w s q x p x y where q and p are variable exponents and s 0 1 and the related nonlocal operator which is a fractional version of the nonhomogeneous p x laplace operator we first give some further qualitative properties of w s q x p x y we also show the strong comparison principle for the fractional p x laplace operator a sub super solution for the nonlocal equations involving the fractional p x laplacian is established

existence and concentration of semiclassical ground state solutions for the generalized chern simons schrodinger system in h1 r2 abstract this paper is concerned with the following singularly perturbed problem in h 1 r 2 e 2 u v x u a 0 u x u j 1 2 a j 2 u x u f u e 1 a 2 u x 2 a 1 u x 1 2 u 2 1 a 1 u x 2 a 2 u x 0 e a 0 u 1 a 2 u 2 2 a 1 u 2 where e is a small parameter v c r 2 r and f c r r by using some new variational and analytic techniques joined with the manifold of pohozaev nehari type we prove that there exists a constant e 0 0 determined by v and f such that for e 0 e 0 the above problem admits a semiclassical ground state solution v e with exponential decay at infinity we also establish a new concentration behaviour of v e as e 0 in particular our results are available to the nonlinearity f u u s 2 u for s 4 6 which extend the existing results concerning the case f u u s 2 u for s 6

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

minimization of quotients with variable exponents abstract let be a bounded domain of r n p c 1 q c and l j n we describe the asymptotic behavior of the minimizers of the rayleigh quotient u l p x u j q x first when j and after when l

computable minque type estimates of variance components abstract the minimum norm quadratic unbiased estimator type minque type of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity the resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates mivque s approaches unity when the true variance ratios are identical and tend to infinity assuming nonnormal effect distributions in the model with two variance components the estimates are asymptotically efficient in a sequence of designs where the number of classes and the number of observations on each class approach infinity it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadrati

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

boundary regularity of minimizers of double phase functionals abstract in this paper we treat the functional of double phase with variable exponents d u g p x a x d u g q x d x where a x is a non negative holder continuous function with 0 1 p x and q x holder continuous functions with 1 p x q x p x and g i j g x i j 1 2 for a continuous positive definite matrix valued function g g we prove that the minimizer of the above functional with suitable dirichlet boundary condition is holder continuous up to the boundary when g is holder continuous we see also that du is locally holder continuous

existence and multiplicity of solutions for dirichlet problem of p x laplacian type without the ambrosetti rabinowitz condition abstract this paper deals with the following variational problem with variable exponents div 1 u p x 1 u 2 p x u p x 2 u f x u in u 0 on where r n n 2 be a bounded domain with the smooth boundary p r is a lipschitz continuous function with 1 p ess inf x p x p x p ess sup x p x n and f c r r is superlinear but does not satisfy the usual ambrosetti rabinowitz type condition under three different superlinear conditions on f at infinity we prove that the above equation has at least a nontrivial solution moreover the existence of infinite many solutions is proved for odd nonlinearity especially some new tricks are introduced to show the boundedness of cerami sequences

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

asymptotic stability for anisotropic kirchhoff systems we study the question of asymptotic stability as time tends to infinity of solutions of dissipative anisotropic kirchhoff systems involving the p x laplacian operator governed by time dependent nonlinear damping forces and strongly nonlinear power like variable potential energies this problem had been considered earlier for potential energies which arise from restoring forces whereas here we allow also the effect of amplifying forces global asymptotic stability can then no longer be expected and should be replaced by local stability the results are further extended to the more delicate problem involving higher order damping terms

finding nicer mappings with lower or equal energy in nonlinear elasticity abstract in the framework of nonlinear elasticity we consider the model energy f u d u x p h det d u x d x where u r n r n with det d u 0 and h 0 0 is convex moreover h t blows up when t 0 we study the problem fix the mapping u with finite energy f u and find a mapping v with the same boundary values with det d v 0 and energy f v not higher than f u such that every component v of v enjoys a kind of maximum and minimum principle due to the constraint det d v 0 a truncation argument does not work on the contrary the constraint det d v 0 makes the problem easy when p n since it is known that every component is weakly monotone so we can take v u in the present work we address the case 2 p n and we give a positive answer if the blow up of h t is of logarithmic type

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

perturbed kirchhoff type neumann problems in orlicz sobolev spaces the aim of this paper is to establish the existence of infinitely many solutions for perturbed kirchhoff type non homogeneous neumann problems involving two parameters to be precise we prove that an appropriate oscillating behaviour of the nonlinear term even under small perturbations ensures the existence of infinitely many solutions our approach is based on recent variational methods for smooth functionals defined on orlicz sobolev spaces

existence of solutions for kirchhoff type problems in musielak orlicz sobolev spaces abstract in this paper we investigate a class of kirchhoff type problem with neumann boundary data in musielak orlicz sobolev spaces using the musielak orlicz theory and mountain pass theorem we establish the existence of nontrivial weak solutions which generalizes the existing results

monte carlo comparison of anova mivque reml and ml estimators of variance components for the one way classification random model with unbalanced data we compare five estimators of 2 a and 2 e the among and within treatments variance components analysis of variance anova maximum likelihood ml restricted maximum likelihood reml and two minimum variance quadratic unbiased mivque estimators mivque 0 is mivque with a priori values 0 and 1 mivque a is mivque with the anova estimates used as a priori s we enforce nonnegativity for all estimators setting any negative estimate to zero in accord with usual practice the estimators are compared through their biases and mse s estimated by monte carlo simulation our results indicate that the anova estimators perform well except with seriously unbalanced data when 2 a 2 e 1 ml is excellent when 2 a 2 e 0 5 and mivque a is adequate further iteration to the reml estimates is unnecessary when 2 a 2 e 1 mivque 0 the default for sass procedure varcomp is poor for estimating 2 a and very poor for 2 e

hodge decomposition of variable exponent spaces of clifford valued functions and applications to dirac and stokes equations in this paper we establish a hodge type decomposition of variable exponent lebesgue spaces of clifford valued functions where one of the subspaces is the space of all monogenic lp x functions using this decomposition we obtain the existence and uniqueness of solutions to the homogeneous a dirac equations with variable growth under certain appropriate conditions and to the stokes equations in the setting of variable exponent spaces of clifford valued functions

renormalized solutions for the fractional p x laplacian equation with l1 data abstract we consider the fractional p x laplacian problem with l 1 data based on the approximation and energy methods we prove the existence and uniqueness of nonnegative renormalized solutions for such problems

a comparison of variance component estimates for arbitrary underlying distributions abstract large sample covariance matrices for the analysis of variance anova minimum norm quadratic unbiased estimator minque restricted maximum likelihood reml and maximum likelihood ml estimates of variance components are presented for the unbalanced one way model when the underlying distributions are not necessarily normal the limiting variances depend on the design sequence on the actual values of the variance components and on the kurtosis parameters of the underlying distributions the skewness parameters and other moments do not affect the limiting distributions because all estimates are consistent and asymptotically normal it is reasonable to compare the estimates using their asymptotic variances thus the efficiency of one estimate relative to another is defined as the ratio of their asymptotic variances and these efficiencies are evaluated numerically and analytically for a variety of nonnormal situations various authors including hocking and kutner 1975 corbeil and searl

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

on bounded entire solutions of some quasilinear elliptic equations abstract in this paper we prove the existence of infinitely many bounded entire solutions ground state entire solutions and entire solutions with prescribed limit at infinity to pdes involving the laplace operator in the setting of orlicz sobolev spaces the main tools used are the sub solution and super solution technique in conjunction with the schauder tychonoff fixed point theorem

the lavrentiev gap phenomenon in nonlinear elasticity the main objective of this paper is to present examples of the lavrentiev phenomenon within the framework of two dimensional nonlinear elasticity loosely speaking this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings we provide a physically natural stored energy density and reasonable though nontraditional boundary conditions such that the energy functional exhibits the lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations the stored energy density w that we produce is smooth materially homogeneous frame indifferent isotropic and polyconvex furthermore the corresponding minimization problem is such that existence of a continuous minimizer follows from known results the basis for our examples is a convex integrand w 0 for which the euler lagrange equations have a very special form we show that the functional associated with this w 0 exhibits the lavrentiev phenomenon for certain problems by making a perturbation to w 0 we create the stored energy density w described in the previous paragraph with other perturbations to the integrand w 0 and modifications of the boundary conditions we are able to produce additional examples of the lavrentiev phenomenon finally we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon

the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous dirichlet boundary condition abstract in this paper we show the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous dirichlet boundary condition under the condition p x 2 n n 2 n 2 3 in particular we improve a priori estimate for weak solutions to the problem with min p x 2 which is an improvement even for constant p

holder continuity of quasiminimizers with nonstandard growth abstract we show the holder continuity of quasiminimizers of the energy functionals f x u u d x with nonstandard growth under the general structure conditions z p x b x y r x g x f x y z z p x b x y r x g x the result is illustrated by showing that weak solutions to a class of a b harmonic equations div a x u u b x u u are quasiminimizers of the variational integral of the above type and thus are holder continuous our results extend work by chiado piat and coscia 1997 fan and zhao 2000 and giaquinta and giusti 1984

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

the fibering map approach to a nonlocal problem involving p x laplacian abstract we study the existence of positive solutions for a p x kirchhoff problem the main tool used is the fibering map approach for the corresponding nehari manifold

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

on quasilinear elliptic problems without the ambrosetti rabinowitz condition abstract using variational arguments we establish the existence of multiple solutions for quasilinear elliptic problems driven by the laplacian operator a major point is that we ensure compactness without the well known ambrosetti rabinowitz superlinearity condition

estimating the gradient of vector fields via div and curl in variable exponent sobolev spaces abstract we show an estimate of the gradient of vector fields u in terms of div u and curl u in vector valued variable exponent sobolev spaces with vanishing normal component on the boundary in simply connected 3d domains

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

existence and blow up rate of large solutions of p x laplacian equations with gradient terms abstract in this paper we investigate boundary blow up solutions of the problem p x u f x u k x u m x in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian our results extend the previous work 25 of y liang q h zhang and c s zhao from the radial case to the non radial setting and 46 due to q h zhang and d motreanu from the assumption that k x u x m x is a small perturbation to the case in which k x u m x is a large perturbation we provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d x and in terms of the growth of the exponents furthermore the comparison principle is no longer applicable in our context since f x is not assumed to be monotone in this paper

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega

global regularity of weak solutions for steady motions of electrorheological fluids in 3d smooth domain abstract in this paper we first show higher integrability of the gradient of weak solutions for steady motions of electrorheological fluids under a further assumption on regularity of external force f in 3d smooth domain based on the higher integrability result we then prove that any weak solution to the problem becomes global strong one under such an assumption on f moreover we improve the lower bound on p x for global regularity of solutions to the problem and their stokes setting even if p x is a constant

confidence intervals on linear combinations of variance components in the unbalanced one way classification a problem that can occur in a variance component analysis is the estimation and construction of confidence intervals for linear combinations of the variance components this article considers the unbalanced one way classification model and develops a procedure that can be used to construct a confidence interval on a linear combination of the among and within group variances a simulation study suggests that the procedure provides an interval that in most cases has an achieved confidence coefficient at least as great as the stated level

existence results for some anisotropic dirichlet problems abstract this paper concerns with a class of elliptic anisotropic dirichlet problems depending of one real parameter on bounded euclidean domains our approach is based on variational and topological methods more concretely along the paper we show the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated pucci and serrin theorem in addition to a local minimum result for differentiable functionals due to ricceri this abstract approach has been developed for equations on carnot groups see 15