structural stability for variable exponent elliptic problems ii the p u laplacian and coupled problems abstract we study well posedness for elliptic problems under the form b u div a x u u f where a satisfies the classical leray lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u a prototype case is the equation u div u p u 2 u f we have to assume that inf x z r p x z is greater than the space dimension n then under mild regularity assumptions on and on the nonlinearities we show that the associated solution operator is an order preserving contraction in l 1 in addition existence analysis for a sample coupled system for unknowns u v involving the p v laplacian of u is carried out coupled elliptic systems with similar structure appear in applications e g in modelling of stationary thermorheological fluids

on the polynomial numerical index of the real spaces c0 l1 and l abstract we give a lower bound for the polynomial numerical index of order k for real lush spaces we use this bound to compute the polynomial numerical index of order 2 of the real spaces c 0 l 1 and l finally we present an example of a real banach space x whose polynomial numerical indices are positive while the ones of its bidual are zero

a nonlinear time compactness result and applications to discretization of degenerate parabolic elliptic pdes abstract we propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems it is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of simon 49 our approach allows to handle various numerical discretizations both in the space variables and in the time variable in particular we can cope quite easily with variable time steps and with multistep time differentiation methods like e g the backward differentiation formula of order 2 bdf2 scheme we illustrate our approach by proving the convergence of a two point flux finite volume in space and bdf2 in time approximation of the porous medium equation

geometry of 2 homogeneous polynomials on lp spaces 1 p abstract we characterise extreme 2 homogeneous polynomials of diagonal form on l p with p 2 and give a class of examples of smooth points we describe the unit ball of p 2 l p 2 for both the cases p 2 and p we apply the results thus found to study the geometry of the predual of p 2 l p 2 and to show that for p 2 an extreme 2 homogeneous polynomial on l p 2 remains extreme when regarded as a polynomial on l p

improvement of eigenvalues of sturm liouville problem with t periodic boundary conditions the computation of eigenvalues of sturm liouville problem with t periodic boundary conditions is considered using the richardson extrapolation based on the finite difference the accuracy of the eigenvalues are improved numerical results demonstrate the usefulness of the correction

geometry of three homogeneous polynomials on real hilbert spaces let h be a two dimensional real hilbert space we give a characterisation of the extreme and the smooth points of the unit ball of the space of three homogeneous polynomials on h in terms of the coefficients of the polynomial we also determine the smooth points of the unit ball of the predual of the space of three homogeneous polynomials

richardson extrapolation of iterated discrete projection methods for eigenvalue approximation in this paper the eigenvalue approximation of a compact integral operator with a smooth kernel is discussed we propose asymptotic error expansions of the iterated discrete galerkin and iterated discrete collocation methods and asymptotic error expansion of approximate eigenvalues we then apply richardson extrapolation to obtain higher order super convergence of eigenvalue approximations numerical examples are presented to illustrate the theoretical estimate

geometry of banach spaces of trinomials abstract for each pair of numbers m n n with m n we consider the norm on r 3 given by a b c m n sup a x m b x n c x 1 1 for every a b c r 3 we investigate some geometrical properties of these norms we provide an explicit formula for m n a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres

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

extreme polynomials and multilinear forms onl1 abstract given a 2 homogeneous polynomialp x y ax2 by2 cxywith real coefficients let p rand p cdenote the norms ofpon the real and complex banach spacel21 respectively we show p r p c and obtain a sufficient and necessary condition on the coefficientsa b andcforpto have norm 1 applying these results we characterize extreme points of the unit ball of p 2l1 for the real banach spacel21and examine them for the complex banach spacel21 we apply them to find extreme points and strongly extreme points of the unit ball of p 2l1 and get an extremal 2 homogeneous polynomial onl1that is not an extreme point we also characterize extreme points and strongly extreme points of the unit ball of l ml1 or l ml1 0 1

iterated fast multiscale galerkin methods for eigen problems of compact integral operators an iterated fast multiscale galerkin method is developed for solving the eigen problem of integral operators with weakly singular kernels we propose a theoretical framework for analysis of the convergence of these methods and show the fast multiscale galerkin method obtain the optimal convergence order for eigenvectors and superconvergence order for eigenvalues while the computational complexity for coefficient matrix is almost optimal the iterated fast multiscale galerkin method can improve the convergence for eigenvectors and exhibit superconvergence through the iteration technique numerical examples are presented to illustrate the theoretical estimates for the error of these methods

on weak solutions of quasilinear parabolic equations sur un domaine borne r u n on considere le probleme aux valeurs limites et initiales parabolique suivant u a u f dans q t o t d x u o t 0 pour m 1 u o dans ou a u est un operateur elliptique quasilineaire d ordre 2m on trouve des conditions suffisantes pour etablir l existence de solutions faibles

the unit ball of the complex mathcal p 3h let h be a two dimensional complex hilbert space and mathcal p 3h the space of 3 homogeneous polynomials on h we give a characterization of the extreme points of its unit ball mathsf b mathcal p 3h from which we deduce that the unit sphere of mathcal p 3h is the disjoint union of the sets of its extreme and smooth points we also show that an extreme point of mathsf b mathcal p 3h remains extreme as considered as an element of mathsf b mathcal l 3h finally we make a few remarks about the geometry of the unit ball of the predual of mathcal p 3h and give a characterization of its smooth points

the spectral expansion for a nonself adjoint hill operator with a locally integrable potential abstract we construct the spectral expansion for the one dimensional schrodinger operator l d 2 dx 2 q x x in l 2 where q x is a 1 periodic lebesgue integrable on 0 1 and complex valued potential we obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator l t for t 0 generated by this operation in l 2 0 1 and the t periodic boundary conditions using it we prove that the eigenfunctions and associated functions of l t form a riesz basis in l 2 0 1 for t 0 then we find the spectral expansion for the operator l

geometry of homogeneous polynomials on two dimensional real hilbert spaces we describe a method which characterises extreme points of the unit ball of the space of homogeneous polynomials of degrees 3 and 4 on two dimensional hilbert spaces and provides classes of examples of extreme and smooth polynomials of an arbitrary degree n

structural stability for variable exponent elliptic problems i the p x laplacian kind problems abstract we study the structural stability i e the continuous dependence on coefficients of solutions of the elliptic problems under the form b u n div a n x u n f n the equation is set in a bounded domain of r n and supplied with the homogeneous dirichlet boundary condition on here b is a non decreasing function on r and a n x n is a family of applications which verifies the classical leray lions hypotheses but with a variable summability exponent p n x 1 p p n p the need for varying p x arises for instance in the numerical analysis of the p x laplacian problem uniqueness and existence for these problems are well understood by now we apply the stability properties to further generalize the existence results the continuous dependence result we prove is valid for weak and for renormalized solutions notice that besides being interesting on its own the renormalized solutions framework also permits us to deduce optimal convergence results for the weak solutions our technique avoids the use of a fixed duality framework like the w 0 1 p x w 1 p x duality and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself in a local or in a non local way the sequel of this paper will be concerned with well posedness of some p u laplacian kind problems and with existence of solutions to elliptic systems with variable solution dependent growth exponent

geometry of spaces of polynomials abstract connections between the shape of the unit ball of a banach space and analytic properties of the banach space have been studied for many years in this article some geometric properties of spaces related ton homogeneous polynomials are considered in particular the rotundity and smoothness of spaces of continuousn homogeneous polynomials and its preduals are studied furthermore an inequality relating the product of the norms of linear functionals on a banach space with the norm of the continuousn homogeneous polynomial determined by the product of the linear functionals is derived this inequality is used to study the strongly exposed points of the predual of the space of continuous 2 homogeneous polynomials

correction of finite element estimates for stum liouville eigenvalues it is shown that a simple asymptotic correction technique of paine de hoog and anderssen reduces the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by the finite element method with linear hat functions and mesh lengthh fromo k 4 h 2 too k h 2 the result still holds when the matrix elements are evaluated by simpson s rule but if the trapezoidal rule is used the error iso k 2 h 2 numerical results demonstrate the usefulness of the correction even for low values ofk

geometric theory of spaces of integral polynomials and symmetric tensor products abstract we investigate the nature of extreme weak exposed and weak strongly exposed points of the unit ball of spaces of n homogeneous integral polynomials on and n fold symmetric products of a banach space e for the space of integral polynomials we show the set of extreme points is contained in the set n e 1 we give smul yan type theorems for spaces of n homogeneous polynomials and n fold symmetric tensors that characterise weak exposed resp weak strongly exposed points in terms of g teaux resp frechet differentiability of the norm on various spaces of tensor products and polynomials our study of the geometry of these spaces has many applications when e has the radon nikod m property we show that the spaces of n homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n when the dimensions of e and n are both at least 2 then the space of n homogeneous polynomials on e is neither smooth nor rotund for a certain class of reflexive banach space the space of n homogeneous approximable polynomials on e is either reflexive or is not isometric to a dual banach space we conclude with a choquet theorem for a space of homogeneous polynomials

equivalent norms in polynomial spaces and applications abstract in this paper equivalence constants between various polynomial norms are calculated as an application we also obtain sharp values of the hardy littlewood constants for 2 homogeneous polynomials on l p 2 spaces 2 p we also provide lower estimates for the hardy littlewood constants for polynomials of higher degrees

spectral corrections for sturm liouville problems abstract the numerical solution of the sturm liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction in this paper we use the shooting method both for eigenvalues and eigenfunctions in integrating the corresponding initial value problems we resort to the boundary value method the technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems a technique to estimate the eigenvalue errors is also suggested and seems to be particularly effective for the higher index eigenvalues numerical experiments on some classical sturm liouville problems are presented