boundary trace embedding theorems for variable exponent sobolev spaces we study boundary trace embedding theorems for variable exponent sobolev space w1 p let be an open bounded or unbounded domain in rn satisfying strong local lipschitz condition under the hypotheses that p l 1 infp x supp x n we prove that there is a continuous boundary trace embedding w1 p lq provided q a measurable function on satisfies condition p x q x n 1 p x n p x for x

on the existence of weak solutions of nonlinear degenerate parabolic system with variable exponents abstract the main goal of the present paper is establishing the existence and uniqueness of weak solutions for the nonlinear degenerate reaction diffusion system with variable exponents a model also is proposed to characterize the invasion of cancer cells towards healthy cells with acidification environment moreover the main results of this paper are obtained using regularization problem the faedo galerkin approximation method some apriori estimates compactness results and the gronwall lemma

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

whitney extension theorems for convex functions of the classes c1 and c1 ω let c be a subset of n not necessarily convex f c r be a function and g c n be a uniformly continuous function with modulus of continuity we provide a necessary and sufficient condition on f g for the existence of a convex function f cc1 n such that f f on c and f g on c with a good control of the modulus of continuity of f in terms of that of g on the other hand assuming that c is compact we also solve a similar problem for the class of c1 convex functions on n with a good control of the lipschitz constants of the extensions namely lip f g finally we give a geometrical application concerning interpolation of compact subsets k of n by boundaries of c1 or c1 1 convex bodies with prescribed outer normals on k

explicit formulas for c 1 1 and c 1 omega _ textrm conv extensions of 1 jets in hilbert and superreflexive spaces abstract given x a hilbert space a modulus of continuity e an arbitrary subset of x and functions f e r g e x we provide necessary and sufficient conditions for the jet f g to admit an extension f f with f x r convex and of class c 1 x by means of a simple explicit formula as a consequence of this result if is linear we show that a variant of this formula provides explicit c 1 1 extensions of general not necessarily convex 1 jets satisfying the usual whitney extension condition with best possible lipschitz constants of the gradients of the extensions finally if x is a superreflexive banach space we establish similar results for the classes c conv 1 x

an extension theorem for convex functions of class c1 1 on hilbert spaces abstract let h be a hilbert space e h be an arbitrary subset and f e r g e h be two functions we give a necessary and sufficient condition on the pair f g for the existence of a convex function f c 1 1 h such that f f and f g on e we also show that if this condition is met f can be taken so that lip f lip g we give a geometrical application of this result concerning interpolation of sets by boundaries of c 1 1 convex bodies in h finally we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous

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

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

on the smoothness of convex envelopes we examine differentiability properties of the convex envelope conv e of a given function e rn cc cc in terms of properties of e it is shown that c1 as well as optimal c1 a regularity results 0 a 1 can be obtained under general conditions

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

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

continuum approach to discreteness we study analytically and numerically continuum models derived on the basis of pade approximations and their effectiveness in modeling spatially discrete systems we not only analyze features of the temporal dynamics that can be captured through these continuum approaches e g shape oscillations radiation effects and trapping but also point out ones that cannot be captured such as peierls nabarro barriers and bloch oscillations we analyze the role of such methods in providing an effective homogenization of spatially discrete as well as of heterogeneous continuum equations finally we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations as well as to compare them with other semicontinuum approximations

sup inf explicit formulas for minimal lipschitz extensions for 1 fields on r n abstract we study the relationship between the lipschitz constant of 1 field introduced in 12 and the lipschitz constant of the gradient canonically associated with this 1 field moreover we produce two explicit formulas which are two extremal minimal lipschitz extensions for 1 fields as a consequence of the previous results for the problem of minimal extension by lipschitz continuous functions from r m to r n we produce explicit formulas similar to those of bauschke and wang see 7 finally we show that wells s extensions see 24 of 1 fields are absolutely minimal lipschitz extensions when the domain of 1 field to expand is finite we provide a counter example showing that this result is false in general

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

bifurcation results for a class of p x laplacian equations abstract in this paper we study the following p x laplacian equation div u p x 2 u u p x 2 u f x u x u x 0 x where r n is bounded 0 under suitable assumptions we obtain the existence of global branches of solutions for the above problem via the subsolution supersolution method

whitney s extension problem for c m let f be a real valued function on a compact set in rn and let m be a positive integer we show how to decide whether f extends to a cm function on rn

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

on smooth extensions of vector valued functions defined on closed subsets of banach spaces let x and z be banach spaces a a closed subset of x and a mapping f a rightarrow z we give necessary and sufficient conditions to obtain a c 1 smooth mapping f x rightarrow z such that f mid a f when either i x and z are hilbert spaces and x is separable or ii x is separable and z is an absolute lipschitz retract or iii x l 2 and z l p with 1 p 2 or iv x l p and z l 2 with 2 p infty where l p is any separable banach space l p s sigma mu with s sigma mu a sigma finite measure space

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

mountain pass and ekeland s principle for eigenvalue problem with variable exponent in this article we study the boundary value problem div u p x 2 u u x 2 u u q x 2 u in u 0 on where is a smooth bounded domain in n and p q are continuous functions on we show that for any 0 there exists infinitely many weak solutions respectively if 0 and small enough then there exists a non negative non trivial weak solution our approach relies on the variable exponent theory of generalized lebesgue sobolev spaces combined with a 2 symmetric version for even functionals of the mountain pass theorem respectively on simple variational arguments based on ekeland s variational principle this article is devoted to the special issue vaxjo conference 2008

multiple solutions for noncooperative p x laplacian equations in rn involving the critical exponent abstract in this paper we study the multiplicity of solutions for a class of noncooperative p x laplacian equations in r n involving the critical exponent by using the limit index theory due to li y q li a limit index theory and its application nonlinear anal 25 1995 1371 1389 and concentration compactness principle we obtain a sequence of radially symmetric solutions

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

variable exponent morrey and campanato spaces abstract the classical morrey spaces and campanato spaces are generalized to the variable exponent case the definitions and some basic properties of the variable exponent morrey and campanato spaces are presented a concept of the p average of the functions is introduced

whitney s extension problems and interpolation of data given a function f e r with e rn we explain how to decide whether f extends to a cm function f on rn if e is finite then one can efficiently compute an f as above whose cm norm has the least possible order of magnitude joint work with b klartag

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

a sharp form of whitney s extension theorem 3 order relations involving multi indices 4 statement of two main lemmas 5 plan of the proof 6 starting the main induction 7 nonmonotonic sets 8 a consequence of the main inductive assumption 9 setup for the main induction 10 applying helly s theorem on convex sets 11 a calder6n zygmund decomposition 12 controlling auxiliary polynomials i 13 controlling auxiliary polynomials ii 14 controlling the main polynomials 15 proof of lemmas 9 1 and 5 2 16 a rescaling lemma 17 proof of lemma 5 3 18 proofs of the theorems 19 a bound for k references