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

removable singularities and non uniqueness of solutions of a singular diffusion equation we prove that the solution u of the equation ut log u u 0 in x0 0 t 2 has removable singularities at x0 0 t if and only if for any 0 0 such that c1 x x0 u x t c2 x x0 holds for all 0 x x0 0 and a t b as a consequence we obtain a sufficient condition for removable singularities at 0 t for solutions of the above equation in 2 0 t and we prove the existence of infinitely many finite mass solutions for the equation in 2 0 t when 0 u0 l1 2 is radially symmetric and u0 lloc1 2

infinitely many non negative solutions for a dirichlet problem involving p x laplacian abstract in this paper we consider a dirichlet problem involving the p x laplacian of the type div u p x 2 u f x u in u 0 on we prove the existence of infinitely many non negative solutions of the problem by applying a general variational principle due to b ricceri and the theory of the variable exponent sobolev spaces

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

renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and l1 data abstract in this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and l 1 data and moreover we obtain the equivalence of renormalized solutions and entropy solutions

on a class of nonlinear singular integral equations with shift on a closed contour in this article we discuss the applicability of an approximation method to a class of nonlinear singular integral equations with carleman shift forming the finite group of iterations preserving or changing orientation

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

lp x ω estimates of vector fields and some applications to magnetostatics problems in the present paper we investigate holder type norm inequalities in terms of div and curl of the vector valued functions in variable exponent lebesgue spaces lp x where r3 moreover by using the obtained results we give some applications for magnetostatics problems

temporal derivatives in the finite element method on continuously deforming grids when solving certain time dependent partial differential equations using a finite element technique on a deforming grid it is shown that there is a need to differentiate the trial solution with respect to the position of each of the moveable node points a result is presented which enables these nonstandard derivatives to be expressed in terms of the standard spatial derivatives of the trial functions provided that certain conditions are satisfied these conditions are investigated and shown to be satisfied for a large class of trial spaces even when the necessary conditions are not satisfied by all of the trial functions being used the result is shown to be still of great use in evaluating these awkward nodal derivatives 1 u x t y u x t a u

a maximum entropy principle for contractive interpolants abstract in this paper a maximum entropy principle is established for a class of continuous contractive m n matrix valued functions g with preassigned fourier coefficients gj j for j 0 1 the function g in this class which achieves the maximum entropy is characterized in a number of different ways and its entropy is computed in terms of a hankel operator based on the prescribed matrices 0 1 a pair of seemingly new szego type formulas are obtained enroute and some applications to nevanlinna pick interpolation are indicated

global attractors for nonlinear parabolic equations with nonstandard growth and irregular data abstract we investigate the large time behavior of solutions to the following nonlinear parabolic equations u t div u p x 2 u f x u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 we first provide the existence and uniqueness of an entropy solution for the problem then through some delicate analysis we establish some regularity results on the solution by which we prove the existence of a global attractor for the solution semigroup

existence and non existence results for nonlinear elliptic equations with nonstandard growth abstract we consider the existence and non existence of solutions for nonlinear elliptic equations whose model is div v p x 2 v v q x 1 v in v 0 on where is a smooth bounded domain in r n n 2 is a bounded radon measure our results are twofold when is absolutely continuous with respect to the p capacity the problem admits a unique solution when is concentrated on a set of zero r capacity r p in then the problem does not admit a solution if q is large enough

on certain minimal entropy extensions appearing in dilation theory i abstract we consider certain natural isometric intertwining dilations which appear in the context of the commutant lifting theorem these results are based on the authors previous work on a strengthened version of the parrott theorem since our extensions are isometric or coisometric the minimize the classical entropy integral

classification and geometrical properties of the xθ valued function spaces abstract this paper is devoted to investigate the x valued function spaces based on the notions of bounded topological lattice banach space net continuous modular net and the dual space net we divide x valued function spaces into two classes norm modular spaces and modular modular spaces for the first class we study the separability of l p i x give a representation of the dual space l p i x find sufficient conditions of the equality l p i x l p i x and prove the reflexivity of l p i x under some reasonable conditions and for the second class we prove the completeness and uniform convexity of l i x in suitable situations to show the naturality and rationality of these results some concrete function spaces such as l p t i l p x t l p t i w 1 p x t and l p t i w 0 1 p x t together with l p p t i l p x t are taken into account

study of the solutions to a model porous medium equation with variable exponent of nonlinearity abstract the authors of this paper study the dirichlet problem of the following equation u t div u x t u f u p x t 1 u the existence and uniqueness of weak solutions are proven also the properties of the solutions are studied which include the property of finite speed of propagation of disturbances localization property and the property of vanishing at a finite time etc

convergence of the finite element method for the porous media equation with variable exponent in this work we study the convergence of the finite element method when applied to the following parabolic equation u t div u gamma mathbf x nabla u f mathbf x t mathbf x in omega subset r m t in 0 t since the problem may be of degenerate type we utilize an approximate problem regularized by introducing a parameter varepsilon we prove under certain conditions on gamma and f that the weak solution of the approximate problem converges to the weak solution of the initial problem when the parameter varepsilon tends to zero discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved finally we present some numerical results of a matlab implementation of the method

existence of solutions for systems arising in electromagnetism abstract in this paper we study the following p x curl systems u p x 2 u a x u p x 2 u f x u g x u u 0 in u p x 2 u n 0 u n 0 on where r 3 is a bounded simply connected domain with a c 1 1 boundary denoted by p 1 is a continuous function a l f g r 3 r 3 are caratheodory functions and are two parameters using variational arguments based on fountain theorem and dual fountain theorem we establish some existence and non existence results for solutions of this problem our main results generalize the results of xiang et al 2017 41 bahrouni and repovs 2018 9 and ge and lu 2019 22

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

arbitrarily many solutions for a perturbed p x laplacian equation involving oscillatory terms this paper deals with a perturbed p x laplacian equation involving oscillatory terms div u p x email protected u u p x 2u q x f u email protected u email protected r n u 0 u x 0as x by using variational methods and the non smooth version symmetric criticality principle we establish a the unperturbed problem p 0 has infinitely many distinct solutions b the number of distinct solutions for p e becomes greater and greater whenever e is smaller and smaller our results are a generalization of the case of laplacian from kristaly to the case of p x laplacian

uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity abstract in this article new properties of variable exponent lebesque and sobolev spaces were examined using these properties we prove that the solution of some parabolic variational inequality is unique with the given conditions

the method of kantorovich majorants to nonlinear singular integral equation with shift the paper is concerned with the applicability of the method of kantorovich majorants to nonlinear singular integral equation with shift the result is illustrated in the generalized holder space

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

existence results of infinitely many solutions for p x kirchhoff type triharmonic operator with navier boundary conditions abstract this paper is concerned with the existence of weak solutions to a class of nonlinear elliptic navier boundary value problem involving the p x kirchhoff type triharmonic operator by means of a variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence of weak solutions for the problem

numerical study of the porous medium equation with absorption variable exponents of nonlinearity and free boundary abstract in this paper we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2d domains with moving boundaries the boundary s movement is governed by an equation prompted by the darcy law and the spatial discretization is defined by a triangulation of the domain at each finite element the solution is approximated by piecewise polynomial functions of degree r 1 using lagrange interpolating polynomials in area coordinates the vertices of the triangles move according to a system of differential equations which is added to the equations of the problem the resulting system is converted into a system of ordinary differential equations in time variable which is solved using a suitable integrator the integrals that arise in the system of ordinary differential equations are calculated using the gaussian quadrature finally we present some numerical results of application of this technique

an obstacle problem and superharmonic functions with nonstandard growth abstract in this article we study solutions and supersolutions of a variable exponent p laplace equation and the corresponding obstacle problem as well as related p superharmonic functions the relationship between these function classes closely parallels the classical case however integrability properties of p superharmonic functions require stronger assumptions

interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents abstract we prove boundary higher integrability for the spatial gradient of very weak solutions of quasilinear parabolic equations of the form u t div a x t u 0 on t t u 0 on t t where the non linear structure a x t u is modeled after the variable exponent p x t laplace operator given by u p x t 2 u to this end we prove that the gradients satisfy a reverse holder inequality near the boundary by constructing a suitable test function which is lipschitz continuous and preserves the boundary values in the interior case such a result was proved in verena bogelein and qifan li 2014 provided p x t p 2 holds and was then extended to the singular case 2 n n 2 p p x t p 2 in qifan li 2017 this restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p 2 and p 2 in this paper we develop a new approach using which we are able to extend the results of verena bogelein and qifan li 2014 qifan li 2017 to the full range 2 n n 2 p p x t p and also obtain analogous results up to the boundary the main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously our techniques improve and simplify many aspects of the method of parabolic lipschitz truncation even in the constant exponent case studied extensively in existing literature to simplify the exposition we will only prove the higher integrability result near the boundary provided the domain satisfies a uniform measure density condition and are non perturbative in nature hence we make no regularity assumptions for the coefficients of the nonlinear operator our techniques are also applicable to higher order equations as well as systems

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

weighted lq estimates for the nonlinear parabolic equations with non standard growth abstract we establish a global calderon zygmund type estimate for the nonlinear parabolic equation in divergence form with variable exponent growth the weighted l q estimates are proved under some precise conditions on the variable exponent the nonlinearity and the boundary of the domain our results extend the existing regularity estimates in variable lebesgue spaces to the weighted case

elliptic equations and systems with nonstandard growth conditions existence uniqueness and localization properties of solutions abstract we study the dirichlet problem for the elliptic equations i d i a i x u d i u p i x 2 d i u c x u u x 2 u f x in a bounded domain r n and the class of elliptic systems j d j a i j x u f i x u i 1 n u u 1 u n satisfying the growth condition x s v r n 2 i j a i j x v v i j a 0 i j v i j p i j x a 0 const 0 the exponents p i j x p i x x are known functions these equations are usually referred to as elliptic equations with nonstandard growth conditions we prove first the theorems of existence of bounded weak solutions and establish sufficient conditions of uniqueness of a weak solution our main purpose is the study of the localization properties of weak solutions we show that the weak solution may identically vanish on a set of nonzero measure in a dead core and derive estimates on the size and location of these dead cores in terms of the problem data the study of the localization properties is performed via the method of local energy estimates

regularity estimates for nonlinear elliptic measure data problems with nonstandard growth abstract we prove a global calderon zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right hand side is a bounded radon measure minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate

infinitely many solutions for a differential inclusion problem in rn involving the p x laplacian abstract in this paper we consider the differential inclusion problem in r n involving the p x laplacian of the type div u p x 2 u u p x 2 u f x u in r n u w 1 p x r n we prove the existence of infinitely many radially symmetric solutions of the problem under suitable hypotheses by applying a nonsmooth variational principle due to marano and motreanu and the theory of the variable exponent sobolev spaces

null controllability of a class of newtonian filtration equations abstract in this paper we study the null controllability of a class of newtonian filtration equations using the properties of finite extinction time and finite speed of propagation we construct a control function in feedback form such that system is exactly null controllable at any time t 0 if we put a nonnegative constraint on the control function this system is not exactly null controllable any more but approximately null controllable for a long time

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

long time behavior for a nonlinear parabolic problem with variable exponents abstract this paper addresses the question of the asymptotic behavior of solutions to the p x laplacian problem u t div u p x 2 u f x u g with general assumptions on f x u and the exponent p x we prove the existence of global attractors in proper spaces then we consider the fractal dimension of global attractors for the problem under suitable conditions we show that the problem admits an infinite dimensional global attractor

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

sharp gradient estimates for quasilinear elliptic equations with p x growth on nonsmooth domains in this paper we study quasilinear elliptic equations with the nonlinearity modelled after the p x p x laplacian on nonsmooth domains and obtain sharp calderon zygmund type estimates in the variable exponent setting in a recent work of 12 the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p x p x see 1 3 and 1 4 here we bridge this gap to obtain the end point case of the estimates obtained in 12 see 1 5 in order to do this we have to obtain significantly improved a priori estimates below p x p x which is the main contribution of this paper we also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

regularity for the porous medium equation with variable exponent the singular case abstract we extend to the singular case the results of e henriques j m urbano intrinsic scaling for pdes with an exponential nonlinearity indiana univ math j 55 5 2006 1701 1721 concerning the regularity of weak solutions of the porous medium equation with variable exponent the method of intrinsic scaling is used to show that local weak solutions are locally continuous

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

renormalized solutions for a nonlinear parabolic equation with variable exponents and l 1 data abstract we prove the well posedness existence and uniqueness of a renormalized solution to nonlinear parabolic equations with variable exponents and l 1 data the functional setting involves lebesgue sobolev spaces with variable exponents

holder estimates for parabolic p x t laplacian systems in this paper we establish the local holder continuity of the spatial gradient of weak solutions to the parabolic p x t laplacian system begin array lll partial t u rm div left a x t du p x t 2 du right 0 end array more precisely we prove that begin array lll du in c rm loc 0 alpha alpha 2 quad rm for some alpha in 0 1 end array provided p and a are holder continuous

asymptotic preserving scheme for highly anisotropic non linear diffusion equations heat transfer in magnetically confined plasmas is a process characterized by non linear and extremely high anisotropic diffusion phenomena standard numerical methods successfully employed in the numerical treatment of classical diffusion problems are generally inefficient or even prone to break down when such high anisotropies come into play leading thus to the need of new numerical techniques suitable for this kind of problems in the present paper the authors propose a numerical scheme based on an asymptotic preserving ap reformulation of this non linear evolution problem generalizing the ideas introduced in a previous paper for the case of elliptic anisotropic problems p degond a lozinski j narski c negulescu an asymptotic preserving method for highly anisotropic elliptic equations based on a micro macro decomposition j comput phys 231 7 2012 2724 2740 the performances of the here proposed ap scheme are tested numerically in particular it is shown that the scheme is capable to deal with problems characterized by a high degree of anisotropy thus proving to be suitable for the study of anisotropic diffusion in magnetically confined plasmas

regularity results of the thin obstacle problem for the p x laplacian abstract we study thin obstacle problems involving the energy functional with p x growth we prove higher integrability and holder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent p x is holder continuous

infinitely many non negative solutions for a p x kirchhoff type problem with dirichlet boundary condition abstract in this paper we consider the dirichlet problem involving the p x kirchhoff type a b 1 p x u p x d x div u p x 2 u f x u in u 0 on we prove the existence of infinitely many non negative solutions of the problem by applying a general variational principle due to b ricceri and the theory of the variable exponent sobolev spaces

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

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

collocation approximation methods for nonlinear singular integrodifferential equations in banach spaces abstract a new approximation method is proposed for the numerical evaluation of the nonlinear singular integrodifferential equations defined in banach spaces the collocation approximation method is therefore applied to the numerical solution of such type of nonlinear equations by using a system of chebyshev functions through the application of the collocation method is investigated the existence of solutions of the system of non linear equations used for the approximation of the nonlinear singular integrodifferential equations which are defined in a complete normed space i e a banach space

interfaces in multidimensional diffusion equations with absorption terms we study the properties of interfaces in solutions of the cauchy problem for the nonlinear degenerate parabolic equation ut um aup in rn 0 t n 1 2 3 with the parameters m 1 p 0 a 0 satisfying the condition m p 2 we show that the velocity of the interface t supp u x t is given by the formula v m m 1 um 1 t where is the solution of the degenerate elliptic equation div u aup 0 0 on t the first term expresses the classical darcy law while the second one models the presence of an external force that makes the interface to move in the inward direction we give explicit formulas which represent the interface t as a bijection from 0 it is proved that the solution u and its interface t are analytic functions of time t and that they preserve the initial regularity in the spatial variables we also show that the regularity of the interface velocity v with respect to the spatial variables is better than it was at the initial instant the analysis is based on a special coordinate transformation a local version of lagrangian coordinates that renders the free boundary stationary

three solutions for a differential inclusion problem involving the p x laplacian abstract in this paper we consider differential inclusion problem involving the p x laplacian of the type div u p x 2 u f x u in u 0 on applying a version of the non smooth three critical points theorem we obtain the existence of three solutions of the problem in w 0 1 p x

lorentz estimates to nonlinear elliptic obstacle problems of p x growth in reifenberg domains abstract we prove a global estimate in the scale of lorentz spaces for the gradient of weak solution to a general nonlinear obstacle problem of p x growth it is mainly assumed that the nonlinearity satisfies small bmo semi norm in the spatial variable and the boundary of domain is flat in the reifenberg sense while the variable exponent p x is a strongly log holder continuous function here an essential ingredient of our main proof is to make use of a combination of the so called large m inequality principle and the geometric argument

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

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

vanishing solutions of anisotropic parabolic equations with variable nonlinearity abstract we study the property of finite time vanishing of solutions of the homogeneous dirichlet problem for the anisotropic parabolic equations u t i 1 n d i a i x t u d i u p i x t 2 d i u c x t u x t 2 u f x t with variable exponents of nonlinearity p i x t x t 1 we show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data if the solution does not identically vanish in a finite time we estimate the rate of vanishing of the solution as t we establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one

newton kantorovich approximations to nonlinear singular integral equation with shift abstract the paper is concerned with the applicability of some new conditions for the convergence of newton kantorovich approximations to solution of nonlinear singular integral equation with shift of uryson type the results are illustrated in generalized holder space

w 1 p regularity for elliptic equations with measurable coefficients in nonsmooth domains we establish global w 1 p estimates for second order elliptic equations in divergence form under the natural assumption that p is log holder continuous to this end we assume that the coefficients are measurable in one variable and have small bmo semi norms in the other variables and the boundary of the domain is reifenberg flat our work is an optimal and natural extension of w 1 p regularity for such equations with merely measurable coefficients beyond lipschitz domains

lorentz estimate with a variable power for parabolic obstacle problems with non standard growths abstract we prove a global lorentz estimate for the variable power of the gradients of weak solution to parabolic obstacle problems with p t x growth over a bounded nonsmooth domain it is mainly assumed that the variable exponents p t x satisfy a strong type log holder continuity the associated nonlinearities are merely measurable in the time variable and have small bmo semi norms in the spatial variables while the underlying domain is quasiconvex

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar