theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

higher integrability of iterated operators on differential forms abstract in this paper we first prove the local higher integrability and higher order imbedding theorems for the iterated operators defined on differential forms then we prove the global higher integrability and higher order imbedding inequalities for these operators finally we demonstrate applications of the main results by examples

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

a caccioppoli inequality and partial regularity in the calculus of variations a new proof is given of the partial regularity of minimisers under the principal assumptions of uniform strict quasiconvexity and polynomial growth the proof is based on a caccioppoli inequality and an indirect blow up argument this avoids the technical complications of a direct argument

norms of the composition of the maximal and projection operators abstract in this paper we develop different norm estimates for the compositions of the maximal and projection operators applied to the solutions of a nonlinear elliptic partial differential equation as applications we also obtain some estimates related to the jacobian and exponents of mappings

a characterization of the weakly continuous polynomials in the method of compensated compactness we present a sufficient condition for weak continuity in the method of compensated compactness the condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two the condition applies to all the classical examples of weakly continuous functions and generalizes the quadratic theorem and the wedge product theorem in fact the condition reduces to the legendre hadamard necessary condition when the polynomial is quadratic and also whenever a certain orthogonality condition is satisfied the condition is de rived by isolating conditions under which the quadratic theorem can be iter ated 1 introduction in this paper we present a pointwise condition sufficient to guarantee weak continuity in the method of compensated compactness our condition which is stated in terms of the wave cone and characteristic set of the first order linear differential constraints in the method generalizes both the wedge product theorem 4 7 and the quadratic theorem 5 6 in a natural way here we restrict our attention to sequences of quadratic maps analogous to se quences of wedge products and corresponding sequences of differential operators analogous to the exterior derivative we consider polynomials obtained by iter ating these quadratic maps our main result is a sufficient condition for the weak continuity of such polynomials on sequences constrained by the differential oper ators we show that the resulting condition is sharp in the sense that it is the best such condition that takes account only of bounds on the derivatives and not more subtle constraints imposed on the oscillations by the polynomial moreover all smooth weakly continuous functions are generated by polynomials cf 1 and every polynomial can be obtained by iterating quadratic maps for instance the rth frechet derivative of an rth order polynomial is a constant r linear form which we can iterate slot by slot thus our condition provides a practical method of testing functions for weak continuity we do not presently know whether all weakly continuous polynomials converge as a result of our condition but it is interesting to note that a necessary condition of murat 3 the best local necessary condition of which we are aware is stated

dirac harmonic equations for differential forms abstract in this paper we introduce the dirac harmonic equation for differential forms and prove some basic norm inequalities including the caccioppoli type inequality poincare type inequality and the weak reverse holder inequality for the solutions of this kind of differential equations

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

riemann hilbert approach to a generalised sine kernel and applications we investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval q q we determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its fredholm determinant further we apply our results to build the resolvent of truncated wiener hopf operators generated by holomorphic symbols finally the leading asymptotics of the fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models

the monotonic property of ls μ averaging domains and weighted weak reverse holder inequality abstract we first prove a local weighted weak reverse holder inequality for a harmonic tensors then we study the monotonic property of newly introduced l s averaging domains which can be viewed as an application of the local weighted reverse holder inequality in l s averaging domains by applying the local result we also obtain some similar results in other kinds of domains

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

global weighted inequalities for operators and harmonic forms on manifolds abstract in this paper both the local and global weighted sobolev poincare imbedding inequalities and poincare inequalities for the composition t g are established where t is the homotopy operator and g is green s operator applied to a harmonic forms on manifolds

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

weighted imbedding theorems in the space of differential forms abstract we prove a r weighted imbedding theorems for differential forms these results can be used to study the weighted norms of the homotopy operator t from the banach space l p d l to the sobolev space w 1 p d l 1 l 1 2 n and to establish the basic weighted l p estimates for differential forms

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

ar λ weighted integral inequalities for a harmonic tensors abstract in this paper we prove the ar weighted caccioppoli type inequality and weak reverse holder inequality for a harmonic tensors we also obtain the ar weighted hardy littlewood inequality for conjugate a harmonic tensors these inequalities can be considered as extensions of the classical results

endpoint strichartz estimates and global solutions for the nonlinear dirac equation abstract we prove endpoint strichartz estimates for the klein gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions as an application global wellposedness of the nonlinear dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable

kinetics of a migration driven aggregation process with birth and death we propose an irreversible aggregation model driven by migration and birth death processes with the symmetric migration rate kernel k k j k ensuremath k j ikj ensuremath upsilon and the birth rate j 1 k and death rate j 2 k proportional to the aggregate s size k based on the mean field theory we investigate the evolution behavior of the system through developing the scaling theory the total mass m 1 is reserved in the j 1 j 2 case and increases exponentially with time in the j 1 g j 2 case in these cases the long time asymptotic behavior of the aggregate size distribution a k t always obeys the scaling law for the ensuremath upsilon l 2 case this model may provide a more natural description for diverse aggregation processes such as the evolution of the distribution of city population and individual wealth

integrability for the jacobian of orientation preserving mappings assume f is a map from into rn where is a bounded domain in rn such that d ln logl s with 0 s 1 i e d n log 1 d s then j l log l 1 s k for any compact subset k where j det d when s 0 we recover a well known result of muller j reine angew math 412 1990 20 34 while the case s 1 was obtained by iwaniec and sbordone arch rational mech anal to appear

poincare inequalities with the radon measure for differential forms we establish the local and global poincare inequalities with the radon measure for the solutions to the nonlinear elliptic partial differential equation for differential forms

lipschitz and bmo norm inequalities for operators abstract in this paper we obtain lipschitz norm inequalities and bmo norm inequalities for operators applied to differential forms satisfying the nonhomogeneous a harmonic equation

inequalities for green s operator with lipschitz and bmo norms in this paper we obtain lipschitz norm and bmo norm inequalities for green s operator applied to differential forms we also establish norm comparison theorems for the bmo norm and the lipschitz norm of green s operator

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

poincare type inequalities with lp logl α norms for green s operator in this paper we first establish local poincare type inequalities with l p logl a norms for green s operator applied to the solutions of the nonhomogeneous a harmonic equation then we extend the local results to global versions in the l f m domain

poincare type inequalities for the homotopy operator with lφ ω norms abstract we obtain poincare type inequalities for the homotopy operator applied to differential forms in l m averaging domains which can be used to estimate the norms of operators and the integral properties of differential forms

weighted integral inequalities for solutions of the a harmonic equation we prove the parametric versions of ar weighted integral inequalities for differential forms satisfying the a harmonic equation these results can be considered as extensions of the classical inequalities for sobolev functions

the riemann hilbert problem associated with the quantum nonlinear schrodinger equation we consider the dynamical correlation functions of the quantum nonlinear schrodinger equation in a previous paper we found that the dynamical correlation functions can be described by the vacuum expectation value of an operator valued fredholm determinant in this paper we show that a riemann hilbert problem can be associated with this fredholm determinant this riemann hilbert problem formulation permits us to write down completely integrable equations for the fredholm determinant and to perform an asymptotic analysis for the correlation function

an example of a quasiconvex function that is not polyconvex in two dimensions we study the different notions of convexity for the function f 2 2 2 det where e 2 2 introduced by dacorogna marcellini we show that f is convex polyconvex quasiconvex rank one convex if and only if 2 3 2 1 1 for some 0 2 3 respectively

determinant representation for correlation functions of spin 1 2 xxx and xxz heisenberg magnets we consider zero temperature correlation functions of the spin 1 2 xxz heisenberg chain in the critical regime 1 1 in a magnetic field starting from the algebraic bethe ansatz we derive representations for various correlation functions in term of determinats of fredholm integral operators

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

dirac structures and their composition on hilbert spaces dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures we study these structures and their properties from an operator theoretic point of view in particular we find necessary and sufficient conditions for the composition of two dirac structures to be a dirac structure and we show that they can be seen as lagrangian hyper maximal neutral subspaces of krein spaces moreover special emphasis is laid on dirac structures associated with operator colligations it turns out that this class of dirac structures is linked to boundary triplets and that this class is closed under composition

two weight imbedding inequalities for solutions to the a harmonic equation abstract we obtain the two weight imbedding inequalities for solutions to the a harmonic equation and explore their applications to the k quasiregular mappings in r n these results can be used to study the properties of the homotopy operator t l p d l w 1 p d l 1 l 1 2 n and to establish the weighted l p estimate for solutions to the a harmonic equation

exact solvability and asymptotic aspects of generalized xx0 spin chains abstract building on our earlier work saeedian et al 2018 we introduce and study generalized xx0 models we explicitly construct a long range interacting spin chain referred to as the selberg model and study the correlation functions of the selberg and xx0 models using a matrix integral representation of the generalized xx0 model and applying asymptotic analysis in non intersecting brownian motion the phase structure of the selberg model is determined we find that tails of the tracy widom distribution of gaussian unitary ensemble govern a discrete to continuous third order phase transition in selberg model the same method also reproduces the gross witten phase transition of the original xx0 model finally we conjecture universal features for the phase structure of the generalized xx0 model

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

imbedding theorems in orlicz sobolev space of differential forms abstract in this paper we prove imbedding inequalities with l norms in the orlicz sobolev space of the forms and establish l norm inequalities for the related operators applied to differential forms we also obtain the global imbedding theorems in l m averaging domains

global poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation both local and global a r weighted poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation in john domains are established as applications of the poincare inequalities for green s operator the a r weighted sobolev imbedding inequalities for green s operator in john domains are also proved

ar ω weighted inequalities for a harmonic tensors and related operators abstract we first prove the local a r weighted poincare inequality for a harmonic tensors on domain and the analogue for the projection operator then we prove the global a r weighted poincare inequality for a harmonic tensors finally using the global a r weighted inequality established for a harmonic tensors we obtain the global a r weighted poincare inequality for the projection operator

estimates of weighted integrals for differential forms in this paper we obtain some weighted integral estimates for differential forms which are generalizations of the poincare inequality the caccioppoli type estimate and the weak reverse holder inequality respectively these results can be used to study the integrability of differential forms and to estimate the integrals for differential forms

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

l φ μ averaging domains and poincare inequalities with orlicz norms abstract we characterize l averaging domains using the whitney covers and the quasihyperbolic metric and study the invariance of l averaging domains under some mappings as applications of the l averaging domains we prove the poincare inequality with orlicz norms for solutions of the non homogeneous a harmonic equation in l averaging domains

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

global estimates for compositions of operators applied to differential forms in this paper we obtain some global l p estimates for compositions of operators applied to differential forms satisfying the a harmonic equation these estimates indicate that the operators tdg and tg are bounded

change of scaling and appearance of scale free size distribution in aggregation kinetics by additive rules the idealized general model of aggregate growth is considered on the basis of the simple additive rules that correspond to a one step aggregation process the two idealized cases were analytically investigated and simulated by the monte carlo method in the desktop grid distributed computing environment to analyze pile up and wall cluster distributions in different aggregation scenarios several aspects of aggregation kinetics change of scaling change of size distribution type and appearance of scale free size distribution driven by the zero cluster size boundary condition were determined by the analysis of evolving cumulative distribution functions the pile up case with a minimum active surface singularity could imitate piling up aggregations of dislocations and the case with a maximum active surface could imitate arrangements of dislocations in walls the change of scaling law for pile ups and walls and availability of scale free distributions for walls were analytically shown and confirmed by scaling fitting moment and bootstrapping analyses of simulated probability density and cumulative distribution functions the initial singular symmetric distribution of pile ups evolves by the infinite diffusive scaling law and later it is replaced by the other semi infinite diffusive scaling law with asymmetric distribution of pile ups in contrast the initial singular symmetric distributions of walls initially evolve by the diffusive scaling law and later it is replaced by the other ballistic linear scaling law with scale free exponential distributions without distinctive peaks the conclusion was made as to possible applications of such an approach for scaling fitting moment and bootstrapping analyses of distributions in simulated and experimental data

exchange driven growth we study a class of growth processes in which clusters evolve via exchange of particles we show that depending on the rate of exchange there are three possibilities i growth clusters grow indefinitely ii gelation all mass is transformed into an infinite gel in a finite time and iii instant gelation in regimes i and ii the cluster size distribution attains a self similar form the large size tail of the scaling distribution is phi x approximately exp x 2 nu where nu is a homogeneity degree of the rate of exchange at the borderline case nu 2 the distribution exhibits a generic algebraic tail phi x approximately x 5 in regime iii the gel nucleates immediately and consumes the entire system for finite systems the gelation time vanishes logarithmically t approximately lnn nu 2 in the large system size limit n infinity the theory is applied to coarsening in the infinite range ising kawasaki model and in electrostatically driven granular layers

higher integrability of green s operator and homotopy operator abstract we prove the higher integrability and higher order imbedding theorems of the composition of homotopy operator and green s operator we also study analogues of the homotopy operator and green s operator respectively

large time and distance asymptotics of field correlation function of impenetrable bosons at finite temperature abstract asymptotics of the temperature time dependent correlation function of two fields one dimensional bosons with infinite point like repulsion is calculated

generalized poincare inequalities for solutions to the a harmonic equation in certain domains abstract we first prove local versions of the poincare inequality for solutions to the a harmonic equation then as applications of the local results we obtain the global versions of the poincare inequality for solutions to the a harmonic equation in l s 0 averaging domains and l s averaging domains respectively these results can be considered as generalizations of the classical poincare theorem

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms