mappings of finite distortion discreteness and openness we establish a sharp integrability condition on the partial derivatives of a mapping with l p integrable distortion for some p n 1 to guarantee discreteness and openness we also show that a mapping with exponentially integrable distortion and integrable jacobian determinant is either constant or both discrete and open we give an example demonstrating the preciseness of our criterion

trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces abstract our aim in this paper is to discuss trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces our result will imply the boundedness of the riesz potential operator from a grand morrey space to a morrey space

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

polyharmonic k hessian equations in rn abstract this work is focused on the study of the nonlinear elliptic higher order equation m u s k u f x r n where the k hessian s k u is the kth elementary symmetric polynomial of eigenvalues of the hessian matrix of the solution and the datum f belongs to a suitable functional space this problem is posed in r n and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m n questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed on the way to construct these proofs some technical results such as a fixed point theorem and a refinement of the critical sobolev embedding which could be of independent interest are introduced

on the operator l f f log f abstract the present note deals with the operator l h 1 r n d r n which takes a given element f of the hardy space to the function f log f in general this function need not be locally integrable nevertheless due to peculiar cancellations of large positive and negative terms in the integral f log f with c 0 r n we are able to give meaning to f log f as a schwartz distribution we find several alternatives for this interpretation of f log f

existence and uniqueness for nonlinear elliptic equations with lower order terms abstract we study the nonlinear dirichlet problem for the elliptic equation div a x u b x u div f in a regular domain r n n 2 u 0 on in hypothesis of lipschitz continuity and strong monotonicity of a we assume that the lower order term b x s verify b x s b x t e x s t for a e x and for any s t r where e is a non negative function in the lorentz space l n q n q without any control on the norm of e with f l p p 1 and q we obtain existence and uniqueness result for distributional solutions u w 1 p whenever p is close to two for q uniqueness results are obtained the main difficulty to solve the problem is due to noncoercivity of the vector field a x s a x b x s moreover no classical structure conditions are satisfied

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

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

embeddings of lorentz type spaces involving weighted integral means we solve the problem of characterizing weights on 0 infinity for which the inequality involving two possibly different general inner weighted means integral infinity 0 integral t 0 f s m2

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

on the optimality of certain sobolev exponents for the weak continuity of determinants keywords optimality of sobolev exponents weak continuity of determinant radon measure reference caa article 1992 005doi 10 1016 0022 1236 92 90071 pview record in web of science record created on 2008 11 25 modified on 2017 05 12

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

finding nicer mappings with lower or equal energy in nonlinear elasticity abstract in the framework of nonlinear elasticity we consider the model energy f u d u x p h det d u x d x where u r n r n with det d u 0 and h 0 0 is convex moreover h t blows up when t 0 we study the problem fix the mapping u with finite energy f u and find a mapping v with the same boundary values with det d v 0 and energy f v not higher than f u such that every component v of v enjoys a kind of maximum and minimum principle due to the constraint det d v 0 a truncation argument does not work on the contrary the constraint det d v 0 makes the problem easy when p n since it is known that every component is weakly monotone so we can take v u in the present work we address the case 2 p n and we give a positive answer if the blow up of h t is of logarithmic type

identification of arma models using intermittent and quantized output observations this paper studies system identification of arma models whose outputs are subject to finite level quantization and random packet dropouts using the maximum likelihood criterion we propose a recursive identification algorithm which we show to be strongly consistent and asymptotically normal we also propose a simple adaptive quantization scheme which asymptotically achieves the minimum parameter estimation error covariance the joint effect of finite level quantization and random packet dropouts on identification accuracy are exactly quantified the theoretical results are verified by simulations

the maximum principle of alexandrov for very weak solutions abstract we extend the classical maximal principle of alexandrov to very weak solutions of the elliptic equation i j 1 n a i j x 2 u x i x j f

fully measurable grand lebesgue spaces abstract using variable exponents we build a new class of rearrangement invariant banach function spaces independent of the variable lebesgue spaces whose function norm is f ess sup x 0 1 p x x f where p x denotes the norm of the lebesgue space of exponent p x assumed measurable and possibly infinite and is measurable too such class contains some known banach spaces of functions among which are the classical and the grand lebesgue spaces and the exp spaces 0 we analyze the function norm and we prove a boundedness result for the hardy littlewood maximal operator via a hardy type inequality

gradient regularity for solutions to quasilinear elliptic equations in the plane we investigate the dirichlet problem diva x v fin v 0on for a quasilinear elliptic equation in a planar domain when f belongs to the zygmund space l logl 12 loglogl 0 1 we prove that the gradient of the variational solution v w01 2 belongs to the space l2 loglogl 2 r2 a main tool is a result on the regularity of the gradient of the solution to the dirichlet problem diva x div in w01 1 where l2 loglogl r2 0 namely if the mapping a r2 r2 satisfies the leray lions type conditions then we prove the estimates l2 loglogl r2 c l2 loglogl r2 by applying a method recently suggested by l greco et al which is based on the uniform estimates l2 r2 c l2 r2 available for 0 provided that l2 r2

note on lusin n condition and the distributional determinant abstract let r n be an open set we show that for a continuous mapping f w 1 n 1 r n with j f l 1 the validity of the lusin n condition implies that the distributional jacobian equals to the pointwise jacobian

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

on the capacity of the gaussian channel with a finite number of input levels an analytic confirmation of an observation made by g ungerboeck 1965 that approximately optimal performance on a gaussian channel with capacity c can be obtained with about 2 sup c 1 levels is given in particular the results imply that by using about 2 sup c 1 levels a rate of nearly c 1 bit can be achieved and that by using 2 sup c 1 levels a rate of about c 0 4 bits can be achieved

grand bochner lebesgue space and its associate space abstract our aim is to introduce the grand bochner lebesgue space in the spirit of iwaniec sbordone spaces also known as grand lebesgue spaces and prove some of its properties we will also deal with the associate space for grand bochner lebesgue spaces

some new results related to lorentz gγ spaces and interpolation abstract we compute the k functional related to some couple of spaces as small or classical lebesgue space or lorentz marcinkiewicz spaces completing the results of 11 this computation allows to determine the interpolation space in the sense of peetre for such couple it happens that the result is always a g space since this last space covers many spaces the motivations of such study are various among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain with data in the space of the integrable function with respect to the distance function to the boundary of

recursive identification of fir systems with binary valued outputs and communication channels this paper investigates the recursive identification of the finite impulse response systems whose outputs are subject to both the binary valued quantization and the communication unreliability the discrete memoryless channel with packet losses and the additive white gaussian noise channel with time delay are considered a stochastic gradient type algorithm and a stochastic approximation type algorithm are respectively proposed to estimate the unknown system parameters under some mild conditions the algorithms can be proved to be convergent in the almost sure and mean square senses also the convergence rates are given numerical examples are included to illustrate the performance of the algorithms

some estimates in gγ p m w spaces abstract we consider a class of rearrangement invariant banach function spaces recently appeared in a paper by the same authors containing at the same time some lorentz spaces classical lebesgue spaces and small lebesgue spaces we discuss the main properties coming directly from the norm and for certain values of the involved parameters we prove some estimates of the norm of the associate space

a new div curl result applications to the homogenization of elliptic systems and to the weak continuity of the jacobian in this paper a new div curl result is established in an open set omega of mathbb r n n geq 2 for the product of two sequences of vector valued functions which are bounded respectively in l p omega n and l q omega n with 1 p 1 q 1 1 n 1 and whose respectively divergence and curl are compact in suitable spaces we also assume that the product converges weakly in w 1 1 omega the key ingredient of the proof is a compactness result for bounded sequences in w 1 q omega based on the imbedding of w 1 q s n 1 into l p s n 1 s n 1 the unit sphere of mathbb r n through a suitable selection of annuli on which the gradients are not too high in the spirit of de giorgi and manfredi the div curl result is applied to the homogenization of equi coercive systems whose coefficients are equi bounded in l rho omega for some rho n 1 over 2 if n 2 or in l 1 omega if n 2 it also allows us to prove a weak continuity result for the jacobian for bounded sequences in w 1 n 1 omega satisfying an alternative assumption to the l infty strong estimate of brezis and nguyen two examples show the sharpness of the results

an improvement of dimension free sobolev imbeddings in r i spaces abstract we prove a dimension invariant imbedding estimate for sobolev spaces of first order into a small lebesgue space and we establish the optimality of its fundamental function namely for any 1 p the inequality f y p 0 1 c p f l p b n f w 0 1 p b n n p where y p 0 1 is a rearrangement invariant banach function space independent of the dimension n b n is the ball in r n of measure 1 and c p is a constant independent of n is satisfied by the small lebesgue space l p p 2 0 1 moreover we show that the smallest space y p 0 1 in the sense of the continuous imbedding such that is true has the fundamental function equivalent to that of l p p 2 0 1 as a byproduct of our results we get that the space l p log l p 2 is optimal in the framework of the orlicz spaces satisfying

bi sobolev mappings with differential matrices in orlicz zygmund classes abstract bi sobolev mappings f r 2 onto r 2 have been defined as those homeomorphisms such that f and f 1 belong to w loc 1 1 we deduce regularity properties of the distortion of f from the regularity of the differential matrix d f 1 and conversely

an extension of guo s theorem via k ψ contractive retractions abstract let x be a infinite dimensional banach space we generalize guo s theorem d j guo eigenvalues and eigenvectors of nonlinear operators chinese ann math 2 1981 65 80 english to k contractions and condensing mappings under a condition which depends on the infimum k of all k 1 for which there exists a k contractive retraction of the closed unit ball of the space x onto its boundary

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

extremal mappings of finite distortion the theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non linear elasticity and the calculus of variations in this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm for instance we show the following suppose that f o is a homeomorphism of the circle with f o 1 in cal w 1 2 2 then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non vanishing jacobian determinant the condition on f o is sharp classically the mapping f o is assumed to be quasisymmetric then there is an extremal quasiconformal mapping with boundary values f o but it is not always unique and it is seldom smooth indeed even when f o is quasisymmetric the cal l 1 minimiser for the distortion function will almost never be quasiconformal we further find that there are many new and unexpected phenomena concerning existence uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex these seem to differ markedly from phenomena observed when studying multi well type functionals

universal spaces for r trees r trees arise naturally in the study of groups of isometries of hyperboiic space an r tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals it follows that an r tree is locally arcwise connected contractible and one dimensional unique and local arcwise connectivity characterize r trees among metric spaces a universal r tree would be of interest in attempting to classify the actions of groups of isometries on r trees it is easy to see that there is no universal r tree however we show that there is a universal separable r tree t 0

calderon zygmund singular operators in extrapolation spaces abstract we study the boundedness of the hardy littlewood maximal operator in abstract extrapolation banach function lattices and their kothe dual spaces the extrapolation spaces are generated by compatible families of banach function lattices on quasi metric measure spaces with doubling measure these results combined with a variant of the integral coifman fefferman inequality imply that every calderon zygmund singular operator is bounded in considered extrapolation spaces we apply these results to extrapolation spaces determined by compatible families of calderon lozanovskii spaces in particular to compatible families of orlicz spaces that are interpolation of weighted l p spaces 1 p with a p weights defined on spaces of homogeneous type

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

covering r trees r free groups and dendrites abstract we prove that every length space x is the orbit space with the quotient metric of an r tree x via a free action of a locally free subgroup x of isometries of x the mapping x x is a kind of generalized covering map called a url map and is universal among url maps onto x x is the unique r tree admitting a url map onto x when x is a complete riemannian manifold m n of dimension n 2 the menger sponge the sierpin ski carpet or gasket x is isometric to the so called universal r tree a c which has valency c 2 0 at each point in these cases and when x is the hawaiian earring h the action of x on x gives examples in addition to those of dunwoody and zastrow that negatively answer a question of j w morgan about group actions on r trees indeed for one length metric on h we obtain precisely zastrow s example

towards identification of wiener systems with the least amount of a priori information on the nonlinearity in this paper we investigate what constitutes the least amount of a priori information on the nonlinearity so that the fir linear part is identifiable in the non gaussian input case three types of a priori information are considered including quadrant information point information and locally monotonous information in all three cases identifiability has been established and corresponding identification algorithms are developed with their convergence proofs

a topological characterization of r trees r trees arise naturally in the study of groups of isometries of hyperbolic space an r tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals r actions on r trees can be viewed as ideal points in the compactification of groups of isometries as such they have applications to the study of hyperbolic manifolds our concern in this paper however is with the topological characterization of r trees our main theorem is the following let x p be a metric space then x is uniquely arcwise connected and locally arcwise connected if and only ijf x admits a compatible metric d such that x d is an r tree essentially we show how to put a convex metric on a uniquely arcwise connected locally arcwise connected metrizable space

stability for p laplace type equation in a borderline case abstract we study the dirichlet problem for a p laplacian type operator in the setting of the orlicz zygmund space q log r n q 1 and 0 more precisely our aim is to establish under which assumptions on 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator

homeomorphism with zero jacobian sharp integrability of the derivative abstract let n 2 we construct a homeomorphism f w 1 1 0 1 n r n such that j f 0 almost everywhere and sup 0 e n 1 e 0 1 n d f n e in particular f w 1 p 0 1 n 0 1 n for all p 1 n

from arzela ascoli to riesz kolmogorov abstract in this paper we study totally bounded sets in banach function spaces bfs from which we characterize compact sets via hausdorff criterion in some non standard function spaces which fall under the umbrella of bfs we obtain a riesz kolmogorov compactness theorem for the grand variable exponent lebesgue spaces

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

mappings of finite distortion hausdorff measure of zero sets we prove that a for a mapping f of finite distortion k in l p n p the n p hausdorff measure of any point preimage is zero provided j f is integrable df in l s with s p and the multiplicity function of f is essentially bounded as a consequence for p n 1 we obtain that the mapping is then open and discrete

regularity of solutions of degenerate a harmonic equations we study the regularity of solutions to degenerate a harmonic equations under suitable integrability assumptions on the ellipticity and growth coefficients in particular we show a self improving property of the gradient of the solutions extending previous results by gehring and iwaniec sbordone valid in the uniformly elliptic setting

a direct approach to the duality of grand and small lebesgue spaces abstract in this paper we show by elementary methods that the quasinorms of the grand and small lebesgue spaces f l p sup 0 t 1 1 log t 1 p t 1 f s p d s 1 p 1 p f l q 0 1 1 log t 1 q 0 t f s q d s 1 q d t t q p p 1 found by fiorenza and karadzhov in a fiorenza g e karadzhov grand and small lebesgue spaces and their analogs z anal anwendungen 23 4 2004 657 681 by using deeply extrapolation interpolation techniques are associate each other in other terms the sharp holder s type inequality 0 1 f g d x c p f l q g l p is proven where the sharpness means that l q is the smallest quasinorm up to equivalences such that the inequality holds the method is based entirely on integral estimates makes use of asymptotic properties of euler s gamma function and gives an explicit estimate of the constant c p all the results are expressed in terms of the more general spaces l p and l q 0

grand lebesgue spaces with respect to measurable functions abstract let 1 p given r n a measurable set of finite lebesgue measure the norm of the grand lebesgue spaces l p is given by f l p sup 0 e p 1 e 1 p e 1 f p e d x 1 p e in this paper we consider the norm f l p obtained replacing e 1 p e by a generic nonnegative measurable function e we find necessary and sufficient conditions on in order to get a functional equivalent to a banach function norm and we determine the interesting class b p of functions with the property that every generalized function norm is equivalent to a function norm built with b p we then define the l p spaces prove some embedding results and conclude with the proof of the generalized hardy inequality

fully measurable small lebesgue spaces abstract we build a new class of banach function spaces whose function norm is p f inf f k 1 f k k 1 ess inf x 0 1 p x x 1 f k where p x denotes the norm of the lebesgue space of exponent p x assumed measurable and possibly infinite constant with respect to the variable of f and is measurable too such class contains some known banach spaces of functions among which are the classical and the small lebesgue spaces and the orlicz space l log l 0 furthermore we prove the following holder type inequality 0 1 f g d t p f p g where p f is the norm of fully measurable grand lebesgue spaces introduced by anatriello and fiorenza in 2 for suitable choices of p x and x it reduces to the classical holder s inequality for the spaces e x p 1 and l log l 0

on polyharmonic regularizations of k hessian equations variational methods abstract this work is devoted to the study of the boundary value problem 1 u 1 k s k u f x r n u n u n 2 u n 1 u 0 x where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f obeys suitable summability properties we prove the existence of at least two solutions of which at least one is isolated strictly by means of variational methods we look for the optimal values of n that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results

recent studies on sobolev mappings abstract a homeomorphism u of the unit disk d r 2 u u 1 u 2 d onto d is a quasiharmonic map if u i w l o c 1 1 i 1 2 are solutions to the system 0 1 div b y u 1 0 a e in d div b y u 2 0 a e in d for a symmetric degenerate elliptic conductivity b b y i e 0 2 2 h y b y h y 2 a e in y d r 2 where h d onto 1 is measurable a sufficient condition that the sobolev homeomorphism u w l o c 1 1 is a quasiharmonic map is that u 1 w loc 1 1 this condition is not necessary because we construct a quasiharmonic map u such that u 1 bv w loc 1 1 see theorem 1 1