homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives in this paper we develop a new discontinuous galerkin dg finite element method for solving time dependent partial differential equations pdes with higher order spatial derivatives unlike the traditional local discontinuous galerkin ldg method the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system stability is ensured by a careful choice of interface numerical fluxes the method can be designed for quite general nonlinear pdes and we prove stability and give error estimates for a few representative classes of pdes up to fifth order numerical examples show that our scheme attains the optimal k 1 th order of accuracy when using piecewise k th degree polynomials under the condition that k 1 is greater than or equal to the order of the equation

estimations de la resolvante pour une molecule diatomique dans l approximation de born oppenheimer making use of an adiabatic operator that takes several electronic states into account we derive a born oppenheimer approximation of the resolvent for a diatomic molecule this is an improvement of a result in kmw1 such a resolvent approximation is useful to obtain an adiabatic approximation of total cross sections see jec2 the strategy we use based on mourre s commutator method and on a new kind of global escape function may be carried over to control the resolvent of some matricial schrodinger operators in the same way we obtain a semiclassical estimate for the resolvent of the semiclassical dirac operator with scalar electric potential extending a result of ce

resonances of polyatomic molecules in the born oppenheimer approximation we study the spectral properties of polyatomic molecules near a scattering level we prove that in the born oppenheimer approximation this study can be reduced to the one of a family of finite matrices of semiclassical pseudodifferential operators more precisely we show that any resonance of the molecule which is close enough to the real axis can be obtained from the discrete spectrum of one of these matrixes

interpolation error based a posteriori error estimation for two point boundary value problems and parabolic equations in one space dimension i derive a posteriori error estimates for two point boundary value problems and parabolic equations in one dimension based on interpolation error estimates the interpolation error estimates are obtained from an extension of the error formula for the lagrange interpolating polynomial in the case of symmetrically spaced interpolation points from this formula pointwise and h 1 seminorm a priori estimates of the interpolation error are derived the interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error these a posteriori error estimates are extended to linear two point boundary problems and parabolic equations computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp adaptive code for solving parabolic systems

existence of infinitely many periodic solutions for ordinary p laplacian systems abstract in this paper we study the existence and multiplicity of non trivial periodic solutions of ordinary p laplacian systems by using the minimax technique in critical point theory we also give an example to illustrate that the obtained results are new even in the case p 2

superconvergence of spectral collocation and p version methods in one dimensional problems superconvergence phenomenon of the legendre spectral collocation method and the p version finite element method is discussed under the one dimensional setting for a class of functions that satisfy a regularity condition m u k l cm k on a bounded domain it is demonstrated both theoretically and numerically that the optimal convergent rate is supergeometric furthermore at proper gaussian points or lobatto points the rate of convergence may gain one or two orders of the polynomial degree

magnetic wkb constructions this paper is devoted to the semiclassical magnetic laplacian until now wkb expansions for the eigenfunctions were only established in the presence of a non zero electric potential here we tackle the pure magnetic case thanks to feynman hellmann type formulas and coherent states decomposition we develop here a magnetic born oppenheimer theory exploiting the multiple scales of the problem we are led to solve an effective eikonal equation in pure magnetic cases and to obtain wkb expansions we also investigate explicit examples for which we can improve our general theorem global wkb expansions quasi optimal agmon estimates and upper bound of the tunelling effect in symmetric cases we also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

homoclinic solutions for ordinary p laplacian systems abstract a new result for the existence of homoclinic orbits is obtained for ordinary p laplacian system d dt u t p 2 u t v t u t f t where p 1 t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function this result generalizes and improves some known results in the previous literature

applications of lobatto polynomials to an adaptive finite element method a posteriori error estimates for hp adaptivity and grid to grid interpolation hp adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids lobatto polynomial based techniques are presented for both an interpolation error based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution the estimates involve high order derivatives of the solution that must be approximated from the computed solution differentiating a taylor like series for error in the lobatto interpolant and using the weak form of the equations yields the correct derivative approximations this leads to a more robust order selection strategy interpolation between grids is done over each element using the lobatto interpolating polynomial explicit formulas for the inverse of the resulting lobatto interpolation matrices are given computational results illustrate the theory

exponentially accurate semiclassical asymptotics of low lying eigenvalues for 2 2 matrix schrodinger operators abstract we consider a simple molecular type quantum system in which the nuclei have one degree of freedom and the electrons have two levels the hamiltonian has the form h 4 2 2 y 2 h y where h y is a 2 2 real symmetric matrix near a local minimum of an electron level e y that is not at a level crossing we construct quasimodes that are exponentially accurate in the square of the born oppenheimer parameter by optimal truncation of the rayleigh schrodinger series that is we construct e and such that o 1 and h e exp 2 where 0

systems of cahn hilliard equations the phase separation of alloys with two or more components is studied with emphasis on more than two components particular attention is given to differences between multicomponent and binary alloys specific topics of the paper include equilibrium theory aspects of the dynamics and numerical simulations in the equilibrium theory it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition unlike the binary theory not all composition variations lead to separation and the compositions are not restricted to the convex hull of the equilibrium concentrations linear analysis is used to predict that a pseudo binary will initially result from spinodal decomposition numerical simulations of the dynamics for a ternary alloy verify this initially but more than two phases often

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

the adiabatic limit of schrodinger operators on fibre bundles we consider schrodinger operators h delta g varepsilon v on a fibre bundle m mathop rightarrow limits pi b with compact fibres and a metric g varepsilon that blows up directions perpendicular to the fibres by a factor varepsilon 1 gg 1 we show that for an eigenvalue lambda of the fibre wise part of h satisfying a local gap condition and every n in mathbb n there exists a subspace of l 2 m that is invariant under h up to errors of order varepsilon n 1 the dynamical and spectral features of h on this subspace can be described by an effective operator on the fibre wise lambda eigenspace bundle mathcal e rightarrow b giving detailed asymptotics for h

anisotropic orlicz sobolev spaces of vector valued functions and lagrange equations abstract in this paper we study some properties of anisotropic orlicz and orlicz sobolev spaces of vector valued functions for a special class of g functions we introduce a variational setting for a class of lagrangian systems we give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on orlicz sobolev space

spectral sequences and adiabatic limits a taylor series analysis of the laplacian as the underlying manifold is demormed leads to a hodge theoretic derivation of the leray spectral sequence

collapsing and the differential form laplacian the case of a smooth limit space we analyze the limit of the p form laplacian under a collapse with bounded sectional curvature and bounded diameter to a smooth limit space as an application we characterize when the p form laplacian has small positive eigenvalues in a collapsing sequence

hp version interior penalty dgfems for the biharmonic equation we construct hp version interior penalty discontinuous galerkin finite element methods dgfems for the biharmonic equation including symmetric and nonsymmetric interior penalty discontinuous galerkin methods and their combinations semisymmetric methods our main concern is to establish the stability and to develop the a priori error analysis of these methods we establish error bounds that are optimal in h and slightly suboptimal in p the theoretical results are confirmed by numerical experiments

on the born oppenheimer expansion for polyatomic molecules we consider the schrodinger operatorp h for a polyatomic molecule in the semiclassical limit where the mass ratioh 2 of electronic to nuclear mass tends to zero we obtain wkb type expansions of eigenvalues and eigenfunctions ofp h to all orders inh this allows to treat the splitting of the ground state energy of a non planar molecule our class of potentials covers the physical case of the coulomb interaction we use methods ofh pseudodifferential operators with operator valued symbols which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials

implicit interpolation error based error estimation for reaction diffusion equations in two space dimensions several authors have proposed an error estimation strategy for the finite element method applied to linear reaction diffusion equations in two space dimensions based on an odd even order dichotomy principle for odd order approximations the estimates are computed directly for even order approximations a second solution is computed although both estimators are asymptotically exact the latter are more robust than the former herein the even order method is extended to all orders greater than one thereby recovering robustness for odd orders proofs of asymptotic exactness are extended to nonlinear reaction diffusion equations computational results demonstrating their effectiveness are presented

aps boundary conditions eta invariants and adiabatic limits we prove an adiabatic limit formula for the eta invariant of a manifold with boundary the eta invariant is defined using the atiyah patodisinger boundary condition and the underlying manifold is fibered over a manifold with boundary our result extends the work of bismut cheeger to manifolds with boundary

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

mountain pass solutions to euler lagrange equations with general anisotropic operator abstract using the mountain pass theorem we show that the problem d d t l v t u t u t l x t u t u t for a e t a b u a u b 0 has a solution in anisotropic orlicz sobolev space we consider lagrangian l f t x v v t x f t x with growth conditions determined by anisotropic g function and some geometric conditions of ambrosetti rabinowitz type

hodge decomposition along the leaves of a riemannian foliation abstract a hodge decomposition theorem is proved for the leafwise laplacian of a riemannian foliation on a closed riemannian manifold

bar partial torsion for complex manifolds and the adiabatic limit torsion for complex manifolds and the adiabatic limit we consider a complex fibration f m l b and pull back bundles e 1 and e 2 over m using the adiabatic limit idea we compute the metric invariant tp e1 tp e2 where tp e denotes the complex ray singer torsion

adiabatic decoupling and time dependent born oppenheimer theory we reconsider the time dependent born oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered the adiabatic decoupling leads to an effective schrodinger equation for the nuclei including contributions from the berry connection

the infinite time limit for the time dependent born oppenheimer approximation we analyze the schrodinger equation i varepsilon 2 tfrac partial partial t psi h varepsilon psi whereh is the hamiltonian of the molecular system consisting of nuclei with masses proportional to 4 and electrons with masses of order 1 using the born oppenheimer method we construct the leading order asymptotic expansion to the exact solutions of the equation we show that if the particles interact through smooth potentials decaying suitably as the distance between particles tends to then the expansion holds uniformly for all timest 0 by similar analysis one can show validity of the expansion fort 0 thus our results hold for scattering theory

some existence results on periodic solutions of euler lagrange equations in an orlicz sobolev space setting abstract in this paper we consider the problem of finding periodic solutions of certain euler lagrange equations we employ the direct method of the calculus of variations i e we obtain solutions minimizing certain functional i we give conditions which ensure that i is finitely defined and differentiable on certain subsets of orlicz sobolev spaces w 1 l associated to an n function we show that in some sense it is necessary for the coercivity that the complementary function of satisfy the 2 condition we conclude by discussing conditions for the existence of minima of i

analytic torsion of hirzebruch surfaces using different forms of the arithmetic riemann roch theorem and the computations of bott chern secondary classes we compute the analytic torsion and the height of hirzebruch surfaces

molecular predissociation resonances near an energy level crossing ii vector field interaction abstract we study the resonances of a two by two semiclassical system of one dimensional schrodinger operators near an energy where the two potentials intersect transversally one of them being bonding and the other one anti bonding assuming that the interaction is a vector field we obtain optimal estimates on the location and on the widths of these resonances

morse type inequalities for dynamical systems and the witten laplacian we provide an analytic proof of morse type inequalities for vector flelds determining a morse decomposition with normally hyperbolic dynamics in the demonstration we reduce the problem to the gradient case using a morse bott lyapunov function and

molecular dynamics at an energy level crossing abstract this paper is a continuation of a previous work 2 about the study of the survival probability modelizing the molecular predissociation in the born oppenheimer framework here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels one of these two levels being confining while the second dissociates we show that the survival probability associated to a certain initial state is a sum of the usual time dependent exponential contribution and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter we also compute explicitly the main contribution of the remainder

resonances dans l approximation de born oppenheimer i resume on etudie l operateur p h2 x y v x y sur r xn r yp lorsque h tend vers zero dans un cas ou des resonances apparaissent en utilisant la methode de feshbach l etude de p est d abord ramenee a celle d un operateur matriciel sur l2 r xn de terme principal diag h2 j x ou les j x sont les valeurs propres de y v x y sur l2 r yp sous l hypothese que 2 admet un puits ponctuel non degenere et des conditions supplementaires sur 1 il est alors prouve que p possede des resonances ayant un developpement asymptotique reel en h 1 2 proches des valeurs propres de h2 2 x

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

a posteriori error estimates for fourth order elliptic problems we extend the dichotomy principle of babuska and yu math numerica sinica 13 1991 89 math numerica sinica 13 1991 307 and adjerid et al math mod meth appl s 9 1999 261 siam j sci comput 21 1999 728 for estimating the finite element discretization error to fourth order elliptic problems we show how to construct a posteriori error estimates from jumps of the third partial derivatives of the finite element solution when the finite element space consists of piecewise polynomials of odd degree and from the interior residuals for even degree approximations on meshes of square elements these estimates are shown to converge to the true error under mesh refinement we also show that these a posteriori error estimates are asymptotically correct for more general finite element spaces computational results from several examples show that the error estimates are accurate and efficient on rectangular meshes

some existence results on periodic solutions of ordinary p laplacian systems some existence theorems are obtained for periodic solutions of ordinary p laplacian systems by the minimax methods in critical point theory

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

mountain pass type periodic solutions for euler lagrange equations in anisotropic orlicz sobolev space abstract using the mountain pass theorem we establish the existence of periodic solution for euler lagrange equation lagrangian consists of kinetic part an anisotropic g function potential part k w and a forcing term we consider two situations g satisfying 2 2 at infinity and globally we give conditions on the growth of the potential near zero for both situations

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

adiabatic limits of eta and zeta functions of elliptic operators we extend the calculus of adiabatic pseudo differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator constructed from an elliptic family of operators indexed by s1 we show that the regularized values t 0 and t t 0 are smooth functions of t at t 0 and we identify their values at t 0 with the holonomy of the determinant bundle respectively with a residue trace for invertible families of operators the functions t s and t t s are shown to extend smoothly to t 0 for all values of s after normalizing with a gamma factor the zeta function satisfies in the adiabatic limit an identity reminiscent of the riemann zeta function while the eta function converges to the volume of the bismut freed meromorphic family of connection 1 forms

a mathematical approach to the effective hamiltonian in perturbed periodic problems we describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective hamiltonians in solid state physics we study the partial differential operators of the formp p hy y d y a hy onr n whenh 0 is small enough wherep x y is elliptic periodic iny with respect to some lattice and admits smooth bounded coefficients in x y a x is a magnetic potential with bounded derivatives we show that the spectral study ofp near any fixed energy level can be reduced to the study of a finite system ofh pseudodifferential operatorse x hd x h acting on some hilbert space depending on we then apply it to the study of the schrodinger operator when the electric potential is periodic and to some quasiperiodic potentials with vanishing magnetic field

dirichlet problems for fully anisotropic elliptic equations the existence of a non trivial bounded solution to the dirichlet problem is established for a class of nonlinear elliptic equations involving a fully anisotropic partial differential operator the relevant operator depends on the gradient of the unknown through the differential of a general convex function this function need not be radial nor have a polynomial type growth besides providing genuinely new conclusions our result recovers and embraces in a unified framework several contributions in the existing literature and augments them in various special instances

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

clarke duality for hamiltonian systems with nonstandard growth abstract we consider the existence of periodic solutions to hamiltonian systems with growth conditions involving g function we introduce the notion of symplectic g function and provide relation for the growth of hamiltonian in terms of certain constant c g associated to symplectic g function g we discuss an optimality of this constant for some special cases we also provide applications to the laplacian type systems

vanishing of cohomology groups and large eigenvalues of the laplacian on p forms for collapsing of closed riemannian manifolds the first positive eigenvalue of the laplacian on p forms may or may not tend to infinity in special cases we show that the existence of the first positive eigenvalue tending to infinity is related to vanishing of cohomology groups of generic fibers

phase space tunneling in multistate scattering abstract we prove exponentially small estimates on the off diagonal terms of the scattering matrix associated to two state semiclassical schrodinger hamiltonians our method is based on phase space tunneling estimates and a splitting of the operator by means of toeplitz type phase space cutoff operators

a posteriori error estimation for hp adaptivity for fourth order equations a posteriori error estimates developed to drive hp adaptivity for second order reaction diffusion equations are extended to fourth order equations a c 1 hierarchical finite element basis is constructed from hermite lobatto polynomials a priori estimates of the error in several norms for both the interpolant and finite element solution are derived in the latter case this requires a generalization of the well known aubin nitsche technique to time dependent fourth order equations we show that the finite element solution and corresponding hermite lobatto interpolant are asymptotically equivalent a posteriori error estimators based on this equivalence for solutions at two orders are presented both are shown to be asymptotically exact on grids of uniform order these estimators can be used to control various adaptive strategies computational results for linear steady state and time dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings