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

application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

transformations of some induced osp 3 2 modules in an so 3 sp 2 basis the modules of the orthosymplectic lie superalgebra osp 3 2 induced from finite dimensional irreducible submodules of the stability subalgebra so 3 gl 1 are investigated the corresponding infinite dimensional irreducible or indecomposable modules the kac modules and the related typical and atypical modules are studied in detail every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so 3 sp 2 for each of these infinite dimensional or finite dimensional irreducible or indecomposable modules relations are written down giving the transformations of the basis under the action of the algebra generators

projective duality of toric manifolds and defect polytopes non singular toric embeddings with dual defect are classified the associated polytopes called defect polytopes are proven to be the class of delzant integral polytopes for which a combinatorial invariant vanishes the structure of a defect polytope is described

renormalized solutions for nonlinear elliptic equations with variable exponents and l1 data abstract we prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and l 1 data the functional setting involves lebesgue sobolev space with variable exponents w 1 p

existence of infinitely many solutions for a neumann problem involving the p x laplacian abstract in this paper we consider the neumann problem involving the p x laplacian of the type div u p x 2 u x u p x 2 u f x u g x u in u 0 on we prove the existence of infinitely many solutions of the problem under weaker hypotheses by applying a variational principle due to b ricceri and the theory of the variable exponent sobolev spaces our results are an improvement and generalization of the relative results obtained by b ricceri for the p laplacian case

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

representations of osp m 2n and young supertableaux young supertableaux are introduced for finite dimensional representations of orthosymplectic superalgebras and used to study their content

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

existence and multiplicity results for a new p x kirchhoff problem abstract in this work we study the existence and multiplicity results for the following nonlocal p x kirchhoff problem 0 1 a b 1 p x u p x d x d i v u p x 2 u u p x 2 u g x u in u 0 on where a b 0 are constants r n is a bounded smooth domain p c with n p x 1 is a real parameter and g is a continuous function the analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties

singular solutions of the p laplace equation soit un sous ensemble ouvert de r n n 1 et p un nombre reel plus grand que 1 on suppose que 0 o et u est p harmonique dans on etudie le comportement de la solution faible u pres de o de l equation apu div u p 2 u 0

on superlinear p x laplacian equations in rn abstract we consider the p x laplacian equations in r n the nonlinearity is superlinear but does not satisfy the ambrosetti rabinowitz type condition we obtain ground states of the equations improving a recent result of fan x l fan p x laplacian equations in r n with periodic data and nonperiodic perturbations j math anal appl 341 2008 103 119 we also establish a bartsch wang type compact embedding theorem for variable exponent spaces then a multiplicity result for the equations is proved for odd nonlinearity

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and l1 data abstract we prove the existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and l 1 data the functional setting involves sobolev spaces with variable exponents w 1 p

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

remarks on ricceri s variational principle and applications to the p x laplacian equations abstract in this paper we give some remarks on a variational principle of ricceri in the case of sequentially weakly lower semi continuous functionals defined on a reflexive real banach space in particular we introduce the notions of ricceri block and ricceri box which are more convenient in some applications than the weakly connected components using the variational principle of ricceri and a local mountain pass lemma we study the multiplicity of solutions of the p x laplacian equations with neumann dirichlet or no flux boundary condition and under appropriate hypotheses in which the integral functionals need not satisfy the ps condition on the global space we prove that the problem has at least seven solutions

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

renormalized solutions of nonlinear elliptic problems in generalized orlicz spaces abstract we study a general class of nonlinear elliptic problems associated with the differential inclusion x u div a x u f u f where f l 1 the vector field a is monotone in the second variable and satisfies a non standard growth condition described by an x dependent convex function that generalizes both l p x and classical orlicz settings using truncation techniques and a generalized minty method in the functional setting of non reflexive spaces we prove existence of renormalized solutions for general l 1 data under an additional strict monotonicity assumption uniqueness of the renormalized solution is established sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem

finite dimensional representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis ii nontypical representations in a series of two papers all finite dimensional irreducible representations of the special linear lie superalgebra sl 1 3 are constructed explicit formulas for the generators in an orthonormal gel fand zetlin basis of the even subalgebra gl 3 are given this paper develops a background for constructing the representations expressions for the transformation properties of the basis under the action of the generators are written down within all typical sl 1 3 modules

algebro geometric characterization of cayley polytopes abstract in this paper we give an algebro geometric characterization of cayley polytopes as a special case we also characterize lattice polytopes with lattice width one by using seshadri constants

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

on the composition factors of kac modules for the lie superalgebras sl m n in the classification of finite dimensional modules of lie superalgebras kac distinguished between typical and atypical modules kac introduced an induced module the so called kac module v with highest weight which was shown to be simple if is a typical highest weight if is an atypical highest weight the kac module is indecomposable and the simple module v can be identified with a quotient module of v in the present paper the problem of determining the composition factors of the kac modules for the lie superalgebra sl m n is considered an algorithm is given to determine all these composition factors and conversely an algorithm is given to determine all the kac modules containing a given simple module as a composition factor the two algorithms are presented in the form of conjectures and illustrated by means of detailed examples strong evidence in support of the conjectures is provided the combinatorial way in which the two algorithms are intertwined is both surprising and interesting and is a convincing argument in favor of the solution to the composition factor problem presented here

chern numbers of ample vector bundles on toric surfaces this article shows a number of strong inequalities that hold for the chern numbers c 1 2 c 2 of any ample vector bundle epsilon of rank r on a smooth toric projective surface s whose topologi

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

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

finite dimensional representations of the lie superalgebra gl 2 2 in a gl 2 gl 2 basis ii nontypical representations all finite dimensional irreducible representations of the general linear lie superalgebra gl 2 2 are written down in matrix form the basis within each representation space is chosen in such a way that it makes evident the decomposition of gl 2 2 into irreducible representations of its even subalgebra gl 2 gl 2 special attention is devoted to the analysis of all nontypical representations and some indecomposible representations

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

on the discrete boundary value problem for anisotropic equation abstract in this paper we consider the discrete anisotropic boundary value problem using critical point theory firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions

atypical representations for type i lie superalgebras a new modified induced module construction is presented for all finite dimensional irreducible typical and atypical modules of a type i basic classical lie superalgebra the method is illustrated with some low dimensional representations of gl m mod n and all representations of gl 2 mod 1

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

on unitary lie superalgebras from the spin orbit supersymmetrisation procedure from the basis elements of a clifford algebra cl n the authors generate a grading leading to a unitary lie superalgebra when n is even such a construction is motivated by the understanding of the specific properties of the fermionic variables in the so called spin orbit coupling procedure of supersymmetrisation in n 2 supersymmetric quantum mechanics the n odd case is also considered and some specific examples are discussed

classifying smooth lattice polytopes via toric fibrations we show that any smooth q normal lattice polytope p of dimension n and degree d is a strict cayley polytope if n 2d 1 this gives a sharp answer for this class of polytopes to a question raised by v v batyrev and b nill

on a dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div v z 1 x u v z n x u l u x u f u k x u u w 0 1 p x in a bounded domain and with nonconvex nonlinearity satisfying some local growth conditions the conditions upon v and l allow for considering the p x laplacian equation we use the relations between critical points and critical values to the primal and a suitable dual action functional to get the existence stability and some properties of the solutions

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

entropy solutions for a doubly nonlinear elliptic problem with variable exponent abstract we study the boundary value problem b u div a x u f in u 0 on where is a smooth bounded domain in r n n 3 and div a x u is a p x laplace type operator with p 1 a measurable function and b a continuous and nondecreasing function from r r we prove the existence and uniqueness of an entropy solution for l 1 data f

representations of the superalgebra f4 and young supertableaux the authors present a method for constructing typical as well as atypical finite dimensional representations of the superalgebra f4 for such a purpose young supertableaux are introduced