developed matrix inequalities via positive multilinear mappings abstract utilizing the notion of positive multilinear mappings we present some matrix inequalities in particular the choi davis jensen inequality f a b f a f b does not hold in general for a matrix convex function f and a positive multilinear mapping we prove this inequality under certain condition on f moreover some kantorovich and convexity type inequalities including positive multilinear mappings are presented

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

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 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

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

γ convergence and homogenization of functionals in sobolev spaces with variable exponents this paper is devoted to homogenization and minimization problems for variational functionals in the framework of sobolev spaces with continuous variable exponents we assume that the sequence of exponents converges in the uniform metric and that the lagrangian has a periodic microstructure then under natural coerciveness assumptions we prove a convergence result and as a consequence the convergence of minimizers solutions to the corresponding euler equations

steady flow of non newtonian fluids monotonicity methods in generalized orlicz spaces abstract our purpose is to show the existence of weak solutions to steady flow of non newtonian incompressible fluids with nonstandard growth conditions of the cauchy stress tensor we are motivated by the fluids of strongly inhomogeneous behavior and characterized by rapid shear thickening since l p framework is not sufficient to capture the described model we describe the growth conditions with the help of a general x dependent convex function and formulate our problem in generalized orlicz spaces

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

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

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

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms the authors study the nonlinear elliptic equation div a x u du div u g x u f x in with the boundary condition u 0 on where is a bounded open subset of rn a u div a x u du is a nonlinear operator of leray lions type from w1 p0 into its dual c0 r n g x t t 0 and f w 1 p no growth hypothesis is assumed on the vector valued function u the term div u may be meaningless for usual weak solutions even as a distribution the authors deal with a weaker form of solutions of which in w1 p0 are called the renormalized solutions of the original problem this weaker form can be formally obtained from by means of pointwise multiplication by h u where h is a c1 function with compact support they prove the existence of renormalized solutions of and give sufficient conditions under which a renormalized solution is a usual weak solution furthermore the authors study ls and l regularity of renormalized solutions

a study of new hypergeometric transformations the aim of this research paper is to establish a quite general transformation involving hypergeometric functions by the method of elementary manipulation of the series representations certain known as well as new hypergeometric transformations and identities not previously recorded in the literature are then deduced by means of the generalized kummer and dixon theorems obtained earlier by lavoie et al an application of a newly obtained identity is also given

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

renyi squashed entanglement discord and relative entropy differences the squashed entanglement quantifies the amount of entanglement in a bipartite quantum state and it satisfies all of the axioms desired for an entanglement measure the quantum discord is a measure of quantum correlations that are different from those due to entanglement what these two measures have in common is that they are both based upon the conditional quantum mutual information in berta et al 2015 j math phys 56 022205 we recently proposed renyi generalizations of the conditional quantum mutual information of a tripartite state on abc with c being the conditioning system which were shown to satisfy some properties that hold for the original quantity such as non negativity duality and monotonicity with respect to local operations on the system b with it being left open to show that the renyi quantity is monotone with respect to local operations on system a here we define a renyi squashed entanglement and a renyi quantum discord based on a renyi conditional quantum mutual information and investigate these quantities in detail taking as a conjecture that the renyi conditional quantum mutual information is monotone with respect to local operations on both systems a and b we prove that the renyi squashed entanglement and the renyi quantum discord satisfy many of the properties of the respective original von neumann entropy based quantities in our prior work berta et al 2015 phys rev a 91 022333 we also detailed a procedure to obtain renyi generalizations of any quantum information measure that is equal to a linear combination of von neumann entropies with coefficients chosen from the set 1 0 1 here we extend this procedure to include differences of relative entropies using the extended procedure and a conjectured monotonicity of the renyi generalizations in the renyi parameter we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy joint convexity of the relative entropy and the holevo bound

jensen s inequality for operators without operator convexity we give jensen s inequality for n tuples of self adjoint operators unital n tuples of positive linear mappings and real valued continuous convex functions with conditions on the bounds of the operators we also study operator quasi arithmetic means under the same conditions

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

approximation of nonlinear wave equations with nonstandard anisotropic growth conditions weak solutions for nonlinear wave equations involving the p x laplacian for p 1 are constructed as appropriate limits of solutions of an implicit finite element discretization of the problem a simple fixed point scheme with appropriate stopping criteria is proposed to conclude convergence for all discretization regularization perturbation and stopping parameters tending to zero computational experiments are included to motivate interesting dynamics such as blowup and asymptotic decay behavior

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

existence of renormalized solutions to elliptic equation in musielak orlicz space abstract we prove existence of renormalized solutions to general nonlinear elliptic equation in musielak orlicz space avoiding growth restrictions namely we consider div a x u f l 1 on a lipschitz bounded domain in r n the growth of the monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose is log holder continuity of m which results in good approximation properties of the space the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

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

renormalized solutions to nonlinear parabolic problems in generalized musielak orlicz spaces abstract we will present the proof of existence of renormalized solutions to a nonlinear parabolic problem t u div a d u f with right hand side f and initial data u 0 in l 1 the growth and coercivity conditions on the monotone vector field a are prescribed by a generalized n function m which is anisotropic and inhomogeneous with respect to the space variable in particular m does not have to satisfy an upper growth bound described by a 2 condition therefore we work with generalized musielak orlicz spaces which are not necessarily reflexive moreover we provide a weak sequential stability result for a more general problem t u div a d u f u f where is a monotone function with respect to the second variable and f is locally lipschitz continuous within the proof we use truncation methods young measure techniques the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive musielak orlicz spaces

pde and ode limit problems for p x laplacian parabolic equations in this work we prove continuity of solutions with respect to initial conditions and parameters and we prove upper semicontinuity of a family of global attractors for problems of the form ut div d u p x 2 u b u in a bounded smooth domain in rn

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

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

transformations of some gauss hypergeometric functions this paper presents explicit algebraic transformations of some gauss hypergeometric functions specifically the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1 k 1 l 1 m such that k l m are positive integers and 1 k 1 l 1 m 1 all algebraic transformations of these gauss hypergeometric functions are considered we show that apart from classical transformations of degree 2 3 4 6 there are several other transformations of degree 6 8 9 10 12 18 24 besides we present an algorithm to compute relevant belyi functions explicitly

computation of highly ramified coverings an almost belyi covering is an algebraic covering of the projective line such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line in general there are 1 dimensional families of these coverings with a fixed ramification pattern that is hurwitz spaces for these coverings are curves in this paper three almost belyi coverings of degrees 11 12 and 20 are explicitly constructed we demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth painleve equation

algebraic transformations of hypergeometric functions and automorphic forms on shimura curves in this paper we will obtain new algebraic transformations of the 2f1hypergeometric functions the main novelty in our approach is the interpretation of identities among 2f1 hypergeometric functions as identities among automorphic forms on different shimura curves

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

concavity of certain matrix trace and norm functions ii abstract we refine epstein s method to prove joint concavity convexity of matrix trace functions of lieb type tr f a p 1 2 b q a p 1 2 and symmetric anti norm functions of the form f a p b q where and are positive linear maps is an operator mean and f x with a certain power is an operator monotone function on 0 moreover the variational method of carlen frank and lieb is extended to general non decreasing convex concave functions on 0 so that we prove joint concavity convexity of more trace functions of lieb type

on the existence and stability of solutions for dirichlet problem with p x laplacian abstract we show the existence and stability of solutions for a family of dirichlet problems div a x u x p x 2 u x b x u x p x 2 u x f u k x u x u x 0 u w 0 1 p x with nonlinearity satisfying some local growth conditions we construct a new duality theory which differs from the known ones in that it does not require a type of a palais smale condition

on matrix inequalities between the power means counterexamples abstract we prove that the known sufficient conditions on the real parameters p q for which the matrix power mean inequality a p b p 2 1 p a q b q 2 1 q holds for every pair of matrices a b 0 are indeed best possible the proof proceeds by constructing 2 2 counterexamples the best possible conditions on p q for which a p 1 p a q 1 q holds for every unital positive linear map and a 0 are also clarified

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

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

a class of nonlinear elliptic problems we obtain a strict coercivity estimate generalizing that of t i seidman j differential equations 19 1975 242 257 in considering spatial variation for second order elliptic operators a u u with radial in the gradient a for rm the estimate is then applied to obtain existence of solutions of boundary value problems a u u u f u u with dirichlet conditions