traces of sobolev functions on fractal type sets and characterization of extension domains abstract we describe traces of sobolev functionsu w1 p r n 1 r nin terms of sobolev spaces on metric spaces 7 our results apply to smooth submanifolds fractal subsets as well as to open subsets of r n in particular if 0 r nis a john domain then we characterize thosew1 p functions which can be extended tow1 p r n if is uniform then this result implies jones extension theorem 14 in the case of traces on fractal subsets our results are related to those of jonsson and wallin 16

noise induced chaos and phase space flux abstract we study the effect of additive noise on near integrable second order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits the noise is represented by a type of shinozuka stochastic process capable of arbitrarily closely approximating gaussian noise with any specified spectrum we derive a formula for the flux factor applicable for any asymptotic mean stationary excitation this derivation shows that to first order the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit it also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux this is true for both random and nonrandom excitations and indicates that for the dynamical systems considered here these two classes of excitation play substantially equivalent roles in the promotion of chaos

plane polynomial automorphisms of fixed multidegree let mathcal g be the group of polynomial automorphisms of the complex affine plane on one hand mathcal g can be endowed with the structure of an infinite dimensional algebraic group see shafarevich in math ussr izv 18 214 226 1982 and on the other hand there is a partition of mathcal g according to the multidegree see friedland and milnor in ergod th dyn syst 9 67 99 1989 let mathcal g d denote the set of automorphisms whose multidegree is equal to d we prove that mathcal g d is a smooth locally closed subset of mathcal g and show some related results we give some applications to the study of the varieties mathcal g m resp mathcal g leq m of automorphisms whose degree is equal to m resp is less than or equal to m

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

elliptic equations with bmo coefficients in reifenberg domains the inhomogeneous dirichlet problems concerning divergence form elliptic equations are studied optimal regularity requirements on the coefficients and domains for the w1 p theory 1 p are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms the domain is a reifenberg domain these conditions for the w1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains in fact these domains might have fractal dimensions 2004 wiley periodicals inc

hessian nilpotent polynomials and the jacobian conjecture let z z 1 z n and let a n i 1 2 z 2 i be the laplace operator the main goal of the paper is to show that the well known jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture for any homogeneous polynomial p z of degree d 4 if m p m z 0 for all m 1 then m p m 1 z 0 when m 0 or equivalently m p m 1 z 0 when m 3 2 3 n 2 1 it is also shown in this paper that the condition m p m z 0 m 1 above is equivalent to the condition that p z is hessian nilpotent i e the hessian matrix hes p z 2 p z i z j is nilpotent the goal is achieved by using the recent breakthrough work of m de bondt a van den essen and various results obtained in this paper on hessian nilpotent polynomials some further results on hessian nilpotent polynomials and the vanishing conjecture above are also derived

images of commuting differential operators of order one with constant leading coefficients abstract we first study some properties of images of commuting differential operators of polynomial algebras of order one with constant leading coefficients we then propose what we call the image conjecture on these differential operators and show that the jacobian conjecture bass et al 1982 bcw van den essen 2000 e de bondt 2009 bo hence also the dixmier conjecture dixmier 1968 d and the vanishing conjecture zhao 2007 z3 of differential operators with constant coefficients are actually equivalent to certain special cases of the image conjecture a connection of the image conjecture and hence also the jacobian conjecture with multidimensional laplace transformations of polynomials is also discussed

a mostly elementary proof of morrey space estimates for elliptic and parabolic equations with vmo coefficients abstract in recent years there has been considerable interest in extending the well known calderon zygmund estimates for the laplacian to more general equations in particular equations with highest order coefficients lying in the sarason space vmo in addition the analogous estimates with morrey spaces replacing lebesgue spaces have been considered these morrey space estimates have been proved by refining the proofs for the l p estimates we shall show that the morrey space estimates follow from the l p estimates via an elementary argument which is very similar to that used by campanato

the conormal derivative problem for elliptic equations with bmo coefficients on reifenberg flat domains we study the inhomogeneous conormal derivative problem for the divergence form elliptic equation assuming that the principal coefficients belong to the bmo space with small bmo semi norms and that the boundary is delta reifenberg flat these conditions for the w 1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domain in fact the reifenberg flatness is the minimal regularity condition for the w 1 p theory

free boundary regularity for harmonic measures and poisson kernels one of the basic aims of this paper is to study the relationship between the geometry of hypersurface like subsets of euclidean space and the properties of the measures they support in this context we show that certain doubling properties of a measure determine the geometry of its support a radon measure is said to be doubling with constant c if c times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center we prove that if the doubling constant of a measure on r n 1 is close to the doubling constant of the ndimensional lebesgue measure then its support is well approximated by ndimensional a ne spaces provided that the support is relatively at to start with primarily we consider sets which are boundaries of domains in r n 1 the n dimensional hausdorfi measure may not be deflned on the boundary of a domain in r n 1 thus we turn our attention to the harmonic measure which is well behaved under minor assumptions see section 3 we obtain a new characterization of locally at domains in terms of the doubling properties of their harmonic measure see section 3 along these lines we investigate how the weak regularity of the poisson kernel of a domain determines the geometry of its boundary sections 5 and 6 pursue this goal as in alt and cafiarelli s work see ac c1 c2 and also jerison s j in both cases the goal is to prove that under the appropriate technical conditions at at points of the boundary the oscillation of the poisson kernel controls the oscillation of the unit normal vector the difierence between our work and the work in ac is that we measure the oscillation in an integral sense bmo estimates

on the image conjecture abstract the image conjecture was formulated by the third author who showed that it implied his vanishing conjecture which is equivalent to the famous jacobian conjecture we prove various cases of the image conjecture and show that how it leads to another fascinating and elusive assertion that we here dub the factorial conjecture various cases of the factorial conjecture are proved

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

l p estimates for parabolic equations in reifenberg domains abstract the inhomogeneous neumann problem for parabolic equations in divergence form is studied an optimal regularity requirement on the domain for the l p theory is investigated assuming that the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms and that the domain is a reifenberg flat domain

caloric measure in parabolic flat domains in this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant the logarithm of the density of caloric measure with respect to a certain projective lebesgue measure is of vanishing mean oscillation we also obtain a partial converse to this result our work generalizes to the heat equation recent work of kenig and toro for the laplacian

control systems with almost periodic excitations for control systems described by ordinary differential equations subject to almost periodic excitations the controllability properties depend on the specific excitation here these properties and in particular control sets and chain control sets are discussed for all excitations in the closure of all time shifts of a given almost periodic function then relations between heteroclinic orbits of an uncontrolled and unperturbed system and controllability for small control ranges and small perturbations are studied using melnikov s method finally a system with two well potential is studied in detail

the strong factorial conjecture abstract in this paper we present an unexpected link between the factorial conjecture 8 and furter s rigidity conjecture 13 the factorial conjecture in dimension m asserts that if a polynomial f in m variables x i over c is such that l f k 0 for all k 1 then f 0 where l is the c linear map from c x 1 x m to c defined by l x 1 l 1 x m l m l 1 l m the rigidity conjecture asserts that a univariate polynomial map a x with complex coefficients of degree at most m 1 such that a x x mod x 2 is equal to x if m consecutive coefficients of the formal inverse for the composition of a x are zero

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

parabolic equations with bmo coefficients in lipschitz domains abstract in this paper we are concerned with certain natural sobolev type estimates for weak solutions of inhomogeneous problems for second order parabolic equations in divergence form the geometric setting is that of time independent cylinders having a space intersection assumed to be locally given by graphs with small lipschitz coefficients the constants of the operator being uniformly parabolic we prove the relevant l p estimates assuming that the coefficients are in parabolic bounded mean oscillation bmo and that their parabolic bmo semi norms are small enough

a nonlinear probabilistic method for predicting vessel capsizing in random beam seas the capsizing of vessels in random beam seas is investigated using a single degree of freedom model which is limited to the roll motion several factors including sea wave spectrum nonlinear righting moment characteristics and nonlinear damping are taken into account in the analysis a nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics specifically the melnikov function and the phase space area flux and random vibrations conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics significant wave height and characteristic wave period and the vessel parameters damping and stiffness coefficients extensive numerical simulations are carried out to demonstrate the validity of the analytical results it is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing

a wiener filter state space flux optimal control against escape from a potential well escape from a potential well often identified with system failure is reformulated for a class of two dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well a control against system failure based on minimizing the state space flux out of this safe region is considered for systems subject to weak external forcing with available control power exceeding a certain threshold the finite lag state space flux optimal control against failure is shown to involve a wiener filter a non wiener solution is proposed for available power below this threshold these results hold for wide sense stationary forcing with any statistics and any spectrum

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

the conormal derivative problem for equations of variational type in nonsmooth domains it is well known that elliptic boundary value problems in smooth domains have smooth solutions but if the domain is say c 1 the solutions need not be lipschitz recently korevaar has identified a class of lipschitz domains in which solutions of the capillary problem are lipschitz assuming the contact angle relates correctly to the geometry of the domain lipschitz bounds for more general boundary value problems in the same class of domains are proved applications to variational inequalities are also considered

on the variety of automorphisms of the affine plane abstract the main subject of our study is ga2 n the variety of automophisms of the affine plane of degree bounded by a positive integern after detailing some definitions and notations in section 1 we give in section 2 an algorithm to decide whether an endomorphism of the affine plane over an integral domain is a tame automorphism then by applying this algorithm to the nagata automorphism we recover easily its known results in section 6 we compute the number of irreducible components of ga2 nwhenn 9 and we show that ga2 nis reducible whenn 4 our proofs are based on a precise decomposition theorem for automorphisms given in section 3 and a characterization of length one automorphisms given in section 5 finally in section 7 we give some details on the casen 4

first passage time of nonlinear ship rolling in narrow band non stationary random seas the generalized extended stochastic central difference gescd method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non stationary random seas the gescd method is based on a combination of the extended stochastic central difference method with a statistical linearization technique modified adaptive time scheme and time coordinate transformation the extended stochastic central difference method is however an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non stationary random disturbances approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined it is concluded that the gescd method is very accurate simple and efficient to apply compared with monte carlo simulation the proposed method is applicable to cases with large nonlinearities and intensive random excitations the approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the monte carlo simulation it is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature

incompatibility of diophantine equations arising from the strong factorial conjecture abstract in the present article we establish new evidence in support of the strong factorial conjecture of edo and van den essen 2 by proving it in several special cases for example we show that the conjecture holds for powers of linear forms and sums of prime powers of the variables finally we show one way of constructing new examples of polynomials satisfying the conjecture using known examples

on the length of polynomial automorphisms of the affine plane the automorphism group of the affine plane is a mysterious and challenging object although we know that it is an amalgamated product of two well known subgroups many questions are still unsolved moreover the group has the structure of an infinite dimensional algebraic group but the interactions between these two structures are not yet clear in this paper we study the length of an element defined using the amalgamated structure with respect to the algebraic structure if the ground field is of characteristic zero we prove that the length is a lower semicontinuous function on the group