on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

linear stability of symplectic maps a general method is presented for analytically calculating linear stability limits for symplectic maps of arbitrary dimension in terms of the coefficients of the characteristic polynomial and the krein signatures explicit results are given for dimensions 4 6 and 8 the codimension and unfolding are calculated for all cases having a double eigenvalue on the unit circle the results are applicable to many physical problems including the restricted three body problem and orbital stability in particle accelerators

convergence of the finite difference scheme for a general class of the spatial segregation of reaction diffusion systems abstract in this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction diffusion systems with m 2 components more precisely we show that the numerical solution u h l given by the difference scheme converges to the l t h component u l when the mesh size h tends to zero provided u l c 2 for every l 1 2 m in particular our proof provides convergence of a difference scheme for the multi phase obstacle problem

random lie group actions on compact manifolds a perturbative analysis a random lie group action on a compact manifold generates a discrete time markov process the main object of this paper is the evaluation of associated birkhoff sums in a regime of weak but sufficiently effective coupling of the randomness this effectiveness is expressed in terms of random lie algebra elements and replaces the transience or furstenberg s irreducibility hypothesis in related problems the birkhoff sum of any given smooth function then turns out to be equal to its integral w r t a unique smooth measure on the manifold up to errors of the order of the coupling constant applications to the theory of products of random matrices and a model of a disordered quantum wire are presented

on the error estimate of finite difference method for the obstacle problem in this paper we derive the finite difference method for the obstacle problem we apply the maximum principle to the obstacle problem in two mesh sets and derive the error estimates we present the numerical results for 5 point second order and 9 point compact fourth order schemes

monotone multigrid methodsfor elliptic variational inequalities ii we derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements the coarse grid corrections are computed from certain obstacle problems the actual constraints are fixed by the preceding nonlinear fine grid smoothing this new approach allows the implementation as a classical v cycle and preserves the usual multigrid efficiency we give 1 o j 3 estimates for the asymptotic convergence rates the numerical results indicate a significant improvement as compared with previous multigrid approaches

local higher integrability for unsteady motion equations of generalized newtonian fluids abstract in this paper we show local higher integrability of the gradient of weak solutions for unsteady motions of generalized newtonian fluids including electrorheological fluids under the restriction p z 3 n 2 n 2 moreover this is a new result even for corresponding stokes setting or a system with p structure depending on the symmetric gradient under the restriction p z 2 n n 2

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

parabolic equations in time dependent reifenberg domains in this paper we define time dependent parabolic reifenberg domains and study lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains the basic assumption is that the principal coefficients are of parabolic bmo space with small parabolic bmo seminorms it is shown that lp estimates hold for time dependent parabolic d reifenberg domains

random dirac operators with time reversal symmetry quasi one dimensional stochastic dirac operators with an odd number of channels time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two this follows by adapting the criteria of guivarch raugi and goldsheid margulis to the analysis of random products of matrices in the group so 2l and then a version of kotani theory for these operators absence of singular spectrum can be shown by adapting an argument of jaksic last if the potential contains random dirac peaks with absolutely continuous distribution

the random phase property and the lyapunov spectrum for disordered multi channel systems a random phase property establishing in the weak coupling limit a link between quasi one dimensional random schrodinger operators and full random matrix theory is advocated briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a markov process with a unique invariant measure which is of geometric nature on the elliptic part of the transfer matrices this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study while the random phase property can up to now only be proved in special models or in a restricted sense we provide strong numerical evidence that it holds in the anderson model of localization a main outcome of the random phase property is a perturbative calculation of the lyapunov exponents which shows that the lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary orthogonal and symplectic ensemble differ by a factor 2 each in an anderson ando model on a tubular geometry with magnetic field and spin orbit coupling the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the lyapunov spectrum

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

l q estimates of gradients for evolutional p laplacian systems abstract we study l q estimates of gradients for evolutional p laplacian systems with discontinuous coefficients and external force given by the divergence of bmo functions the result obtained here may be regarded as a partial generalization of the calderon zygmund theory in l q estimate for the parabolic operator of second order to the evolutional p laplace operator

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

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

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

global gradient estimates for the parabolic p x t laplacian equation abstract in this paper we consider the global gradient estimates for weak solutions of the parabolic p x t laplacian type equation in a reifenberg flat domain

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

l p estimates for parabolic equations in reifenberg domains abstract the inhomogeneous neumann problem for parabolic equations in divergence form is studied an optimal regularity requirement on the domain for the l p theory is investigated assuming that the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms and that the domain is a reifenberg flat domain

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

the weak solutions of an evolutionary p x laplacian equation are controlled by the initial value abstract consider the evolutionary p x laplacian equation with a convection term and a source term u t div a x u p x 2 u b i u x i f u x t a x 0 x a x 0 x the equation is degenerate on the boundary if a 1 p 1 d x c the stability of the weak solutions for the initial boundary value problem is studied if a 1 d x c and p 2 the weak solutions are controlled by the initial value completely

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

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

global higher regularity of solutions to singular p x t parabolic equations abstract we study the homogeneous dirichlet problem for the equation u t div u p x t 2 u f x t u in the cylinder q t 0 t r d d 2 it is assumed that p x t 2 d d 2 2 and p p t are bounded a e in q t we find conditions on p x t f x t u and u x 0 sufficient for the existence of strong solutions local or global in time it is proven that the strong solutions possess the property of global higher regularity u t l 2 q t u l 0 t l 2 d i j 2 u p x t l 1 q t

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t

parabolic equations in reifenberg domains natural sobolev type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder 0 t which is reifenberg flat in the space direction the principal coefficients of the operator are assumed to be in bmo space with their bmo semi norms small enough

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f