the minimal period problem for nonconvex even second order hamiltonian systems abstract in this paper we study the minimal period problem for even autonomous second order hamiltonian systems defined on r nwithout any convexity assumption by using the variational methods we obtain estimates on the minimal period of the corresponding nonconstant periodic solution of the superquadratic and asymptotically linear hamiltonian systems

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

approximation in higher order sobolev spaces and hodge systems abstract let d 2 be an integer 1 l d 1 and be a differential l form on r d with w 1 d coefficients it was proved by bourgain and brezis 5 theorem 5 that there exists a differential l form on r d with coefficients in l w 1 d such that d d in the same work bourgain and brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space w 2 d 2 or more generally in the fractional sobolev spaces w s p with s p d we give a positive answer to this question provided that d l d 1 where is the largest positive integer such that min p d the proof relies on an approximation result interesting in its own right for functions in w s p by functions in w s p l even though w s p does not embed into l in this critical case the proofs rely on some techniques due to bourgain and brezis but the context of higher order and or fractional sobolev spaces creates various difficulties and requires new ideas and methods

a binless correlation measure reduces the variability of memory reactivation estimates summary the standard procedure for measuring correlations between pairs of spike trains is to count the numbers of spikes occurring within a specified set of time intervals partitioning the continuous time line into discrete bins of width w seconds one then computes the pearson correlation between pairs of the bin occupancy vectors this method introduces a form of quantization noise similar to that in analog to digital signal processing devices due to the arbitrary positioning of bin boundaries relative to pairs of spikes small changes in bin width and small uniform shifts in bin boundaries typically produce large variations in the apparent correlation an alternative method of determining a correlation between pairs of spike trains was recently introduced rather than discretize the data in time the original spike trains are convolved with a gaussian kernel with parameter chosen to give an effective width matching the bin width w 12 calculating the pearson correlation of the resulting smooth functions gives an estimate of the correlation between the spike trains matching that given by the bin based procedure without introducing the significant variability of the bin based estimate measures of memory reactivation based on the partial correlations between ensembles of pairwise spike train correlations are biased downwards by the quantization noise present in the pairwise correlation estimates using a binless method to measure pairwise correlation we find that the partial correlation or explained variance of rat hippocampal maze activity and post maze sleep taking into account pre maze sleep correlations increases significantly over estimates made with the standard bin based procedure copyright 2007 john wiley sons ltd

hierarchical construction of bounded solutions in critical regularity spaces we construct uniformly bounded solutions for the equations divu d f and curlu d f in the critical cases f 2 l d t d r and f2 l 3 t 3 r 3 respectively criticality in this context manifests itself by the lack of a linear solution operator mapping l d 7 l 1 t d thus the intriguing aspect here is that although the problems are linear construction of their solutions is not our constructions are special cases of a general framework for solving linear equations of the form lu d f where l is a linear operator densely defined in banach space b with a closed range in a proper subspace of lebesgue space l p and with an injective dual l the solutions are realized in terms of a multiscale hierarchical representation u d p 1 jd1 uj interesting for its own sake here the uj s are constructed recursively as minimizers of ujc1d arg min u kukbc jc1krj luk p lp

an alternating direction method of multipliers for the numerical solution of a fully nonlinear partial differential equation involving the jacobian determinant we consider the dirichlet problem for a partial differential equation involving the jacobian determinant in two dimensions of space the problem consists in finding a vector valued function such that the determinant of its gradient is given pointwise in a bounded domain together with essential boundary conditions this problem was initially considered in dacorogna and moser ann inst h poincare anal non lineaire 7 1990 pp 1 26 and several theoretical generalizations have been derived since in this work we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data the proposed method relies on an augmented lagrangian algorithm with biharmonic regularization and low order mixed finite element approximations an iterative method allows us to decouple the nonlinearity and the differential operators numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems an erratum has bee

dual morse index estimates for periodic solutions of hamiltonian systems in some nonconvex superquadratic cases in l ekeland and hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period for hamiltonian systems assuming that the hamiltonian function h is strictly convex with superquadratic behaviour at the origin and at infinity the arguments given in l are based on an estimate for the morse index of a critical point of mountain pass type for a functional which is associated with the hamiltonian system via the duality principle of clarke and ekeland 2 unfortunately the functional is only c and not c then the result is achieved only through a finite dimensional reduction and as the authors themselves say through an inordinate amount of technicalities in this paper we give a direct definition of the morse index of a periodic solution which does not use the duality principle indeed it is based on a study of the morse indexes of approximating solutions obtained by a galerkin scheme such as the one introduced by rabinowitz in 3 taking into account some results about linking point theory recently stated in 4 71 we get the same index estimates as in l but in a simple way moreover some generalizations are obtained in the sense that the convexity assumption on h can be replaced either by a local convexity condition or by a condition on the second derivative of h plus eventually a symmetry condition see also 8 91 for some other results obtained in this framework in particular in the locally convex case the index estimates yield direct information on the minimal period of the corresponding solution finally let us remark that this definition of the morse index can be adapted to other nonconvex cases and one can still get some index estimates of the same type

nucleation of instability of the meissner state of 3 dimensional superconductors this paper concerns a nonlinear partial differential system in a 3 dimensional domain involving the operator curl2 which is a simplified model used to examine nucleation of instability of the meissner state of a superconductor as the applied magnetic field reaches the superheating field we derive a priori c 2 estimates for a weak solution h the curl of the magnetic potential and determine the location of the maximal points of curlh which correspond to the nucleation of instability of the meissner state we show that if the penetration length is small the solution exhibits a boundary layer if the applied magnetic field is homogeneous curlh is maximal around the points on the boundary where the applied field is tangential to the surface

the form boundedness criterion for the laplacian operator abstract in this paper we characterize the class of measurable functions or more generally real or complex valued distributions f such that the commutator operator c f f acting from h 1 r d to h 1 r d in other words we give a complete solution to the problem for the boundedness of the laplacian which is fundamental to quantum mechanics

the minimal period problem of periodic solutions for autonomous superquadratic second order hamiltonian systems abstract in this paper we study the existence of periodic solutions with prescribed minimal period for superquadratic autonomous second order hamiltonian systems defined on r n with no convexity assumptions we use the direct variational approach for this problem on a w 1 2 space of even functions and prove new iteration inequalities on morse indices using these tools and the saddle point theorem we obtain results under precisely rabinowitz superquadratic condition on potential functions we show that for every t 0 the above mentioned system possesses a t periodic even solution with minimal period not smaller than t n 2

solvability of the divergence equation implies john via poincare inequality abstract let r 2 be a bounded simply connected domain we show that for a fixed every p 1 the divergence equation div v f is solvable in w 0 1 p 2 for every f l 0 p if and only if is a john domain if and only if the weighted poincare inequality u x u q d x c u x q dist x q d x holds for some every q 1 this gives a positive answer to a question raised by russ 2013 in the case of bounded simply connected domains in higher dimensions similar results are proved under some additional assumptions on the domain in question

embeddings for a weakly differentiable functions on domains abstract we prove that the inhomogeneous estimate of vector fields on balls in r n b d k 1 u n n 1 d x n 1 n c b a u u d x for all u c b r n holds if and only if the linear constant coefficient differential operator a of order k has finite dimensional null space fdn this generalizes the gagliardo nirenberg sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full space r n d k 1 u n n 1 d x n 1 n c r n a u d x for all u c c r n r n proved by van schaftingen precisely for elliptic and cancelling ec operators building on fundamental l 1 estimates from the works of bourgain and brezis we prove that fdn strictly implies ec and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level

on the existence of periodic solutions for a class of symetric hamiltonian systems abstract hamiltonian systems are used to model the motion of discrete mechanical systems this paper establishes the existence of periodic solutions for a class of such systems the method developed to prove existence should be useful for other such problems keywords minimax technique ordinary differential equations and variational problem

bourgain brezis inequalities on symmetric spaces of non compact type abstract let m be a global riemannian symmetric space of non compact type we prove a duality estimate for pairings of divergence free l 1 vector fields with vector fields in a critical sobolev space on m m f d v c f l 1 d v l m d v this estimate provides a remedy for the failure of a critical sobolev embedding on such symmetric spaces

an optimal sobolev embedding for l1 abstract in this paper we establish an optimal lorentz space estimate for the riesz potential acting on curl free vectors there is a constant c c d 0 such that i f l d d 1 r d r d c f l 1 r d r d for all fields f l 1 r d r d such that curl f 0 in the sense of distributions this is the best possible estimate on this scale of spaces and completes the picture in the regime p 1 of the well established results for p 1

form boundedness of the general second order differential operator we give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator i j ln aij i j j 1nbj j c with real or complex valued distributional coefficients aij bj and c acting from the sobolev space w1 2 n to its dual w 1 2 n this enables us to obtain analytic criteria for the fundamental notions of relative form boundedness compactness and infinitesimal form boundedness of c with respect to the laplacian on l 2 n in particular we establish a complete characterization of the form boundedness of the schrodinger operator i a 2 q with magnetic vector potential a lloc2 n n and q d n 2006 wiley periodicals inc

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

some results on solutions of minimal period to superquadratic hamiltonian systems then it is well known that rabinowitz s theorem see 8 guarantees the existence for every t 0 of at least one nontrivial solution to l but he left open the question to find a solution having minimal period t when h is convex some results have been recently obtained about such a problem via duality methods the problem has been completely solved by ambrosetti mancini see 2 7 with a suitable hypothesis on the second derivative of h and by clarke ekeland see 4 replacing hj ha and h5 with a subquadratic hypothesis in another framework rabinowitz determines in 9 a family of solutions to l having minimal periods which diverge in the superquadratic case see theorem 2 34 under the

a new approach to counterexamples to l 1 estimates korn s inequality geometric rigidity and regularity for gradients of separately convex functions the derivation of counterexamples to l1 estimates can be reduced to a geometric decomposition procedure along rank one lines in matrix space we illustrate this concept in two concrete applications firstly we recover a celebrated and rather complex counterexample by ornstein proving the failure of korn s inequality and of the corresponding geometrically nonlinear rigidity result in l1 secondly we construct a function f 2 which is separately convex but whose gradient is not in bvloc in the sense that the mixed derivative 2f x1 x2 is not a bounded measure

a limiting case for the divergence equation we consider the equation text div mathbb y f with f a zero average function on the torus mathbb t d in their seminal paper bourgain and brezis j am math soc 16 2 393 426 2003 electronic proved the existence of a solution mathbb y in w 1 d cap l infty for a datum f in l d we extend their result to the critical sobolev spaces w s p with s 1 p d and p ge 2 more generally we prove a similar result in the scale of triebel lizorkin spaces we also consider the equation text div mathbb y f in a bounded domain varomega subject to zero dirichlet boundary condition

gagliardo nirenberg inequalities for differential forms in heisenberg groups the l 1 sobolev inequality states that for compactly supported functions u on the euclidean n space the l n n 1 norm of a compactly supported function is controlled by the l 1 norm of its gradient the generalization to differential forms due to lanzani and stein and bourgain and brezis is recent and states that a the l n n 1 norm of a compactly supported differential h form is controlled by the l 1 norm of its exterior differential du and its exterior codifferential delta u in special cases the l 1 norm must be replaced by the mathcal h 1 hardy norm we shall extend this result to heisenberg groups in the framework of an appropriate complex of differential forms

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

multiple periodic solutions for γ symmetric newtonian systems abstract the existence of periodic solutions in symmetric newtonian systems x f x can be effectively studied by means of the o 2 equivariant gradient degree with values in the euler ring u o 2 in this paper we show that in the case of being a finite group the euler ring u o 2 and the related basic degrees are effectively computable using euler ring homomorphisms the burnside ring a o 2 and the reduced o 2 degree with no free parameters we present several examples of newtonian systems with various symmetries for which we show existence of multiple periodic solutions we also provide exact value of the equivariant topological invariant for those problems

solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem on considere le systeme hamiltonien jz h z ou j x y y x x y r n r n h est non convexe a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l exposant dans ce cas on demontre que pour t 0 l existence de solutions pour le systeme hamiltonien de periode minimale t

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

infinitesimal form boundedness and trudinger s subordination for the schrodinger operator we give explicit analytic criteria for two problems associated with the schrodinger operator h q on l2 n where q d n is an arbitrary real or complex valued potential

on a class of ill posed minimization problems in image processing in this paper we show that minimization problems involving sublinear regularizing terms are ill posed in general although numerical experiments in image processing give very good results the energies studied here are inspired by image restoration and image decomposition rewriting the nonconvex sublinear regularizing terms as weighted total variations we give a new approach to perform minimization via the well known chambolle s algorithm the approach developed here provides an alternative to the well known half quadratic minimization one

lifting degree and distributional jacobian revisited let g i 0 1 s1 if g vmo we may write g ei for some vmo this is unique modulo 2 see 13 and the earlier work 14 there is no control of bmo in terms of g bmo since we always have g bmo 2 and bmo can be arbitrarily large recall however that when g bmo is sufficiently small there is a linear estimate bmo c g bmo see 13 theorem 4 14 and remark 0 2 below we are going to establish that a norm slightly stronger than g bmo does control bmo consider for 1 p 0 s 1 the fractional sobolev space ws p i equipped with its standard seminorm

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

distributions for which div v f has a continuous solution the equation div upsilon f has a continuous weak solution in an open set u subset of r m if and only if the distribution f satisfies the following condition the f phi i converge to 0 for every sequence phi i of test functions such that the support of each i is contained in a fixed compact subset of u and in the l 1 norm phi i converges to 0 and del phi i is bounded c 2007 wiley periodicals inc

essential critical points of linking type and solutions of minimal period to superquadratic hamiltonian systems in a previous paper 3 the authors stated a result about the existence of t periodic solutions for any t 0 of autonomous hamiltonian systems having minimal period t or t 2 in the case where the hamiltonian function h has a superquadratic behaviour of the type considered by ekeland and hofer in 2 but without a global convexity assumption which is replaced by a local convexity condition in this paper we use the notion of essentiality for critical points of linking type see i 4 71 for an exhaustive exposition in order to construct a t periodic solution having t as its minimal period we remark that this result really generalizes the main theorem by ekeland and hofer in 2 the solution is found through a riesz galerkin method as exhibited in 3 following the basic scheme introduced in 6 by rabinowitz and the use of the marino prodi perturbation method more precisely as a first step one finds a critical point of linking type 2 for the restriction f of f to a suitable finite dimensional space for any n e n then some suitable estimates for derived by the linking construction enable the use of the marino prodi theorem in order to approximate each 2 by a sequence zig in such a way that z is a nondegenerate critical point of a functional fim with ff f in the c2 norm as m oo indeed 2 can be chosen as a critical point of linking type satisfying the same estimates as 2 and verifying the further property of essentiality roughly speaking one says that a linking critical point of a functional is not essential or avoidable if by looking at its linking definition one can find a local deformation i e in the neighbourhood of w which enables the construction for any minimizing sequence of surfaces having infinitesimal distances from w of another minimizing sequence of surfaces which keep away from w the solution z of the hamiltonian system is found as the limit of a suitable diagonal subsequence 1 zi j the fact that the minimal period of z can be either t or t 2 derives from some arguments which were already exploited in 3 based on some estimates for the morse indexes of finite dimensional linking points see 7 as a final step one shows that t 2 cannot be a period for z hence the conclusion that t is the minimal period of z by showing that the property of t 2 periodicity for z should yield the avoidability of the points of a subsequence of 1 first of all one looks at the maximum negative eigenvalue uj of d2f zj