on homoclinic orbits for a class of noncoercive superquadratic hamiltonian systems abstract in this paper we prove the existence of homoclinic solutions for a class of noncoercive first order hamiltonian systems j x m t x u g t u x 0 by the minimax methods in critical point theory specially a generalized mountain pass theorem when u is a linear operator with adjoint u and g t y satisfies the superquadratic condition g t x y 2 as y uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

homoclinic orbits on compact hypersufaces in 293 1293 1293 1 of restricted contact type consider a smooth hamiltonian system in 2n dot x jh x the energy surface x h x h 0 being compact and 0 being a hyperbolic equilibrium we assume moreover that 0 is of restricted contact type these conditions are symplectically invariant by a variational method we prove the existence of an orbit homoclinic i e non constant and doubly asymptotic to 0

periodic solutions for a class of first order super quadratic hamiltonian system abstract based on a developed local linking theorem by replacing the palais smale condition with a cerami one we consider the following unbounded hamiltonian system 0 1 t u x u v x u h v t x u v t v x v v x v h u t x u v for t x r where r n n 1 is a bounded domain with smooth boundary the nonlinearities are assumed to be super quadratic at the infinite which are different from those investigated previously

a priori error estimates for three dimensional parabolic optimal control problems with pointwise control in this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise dirac type control in space on three dimensional domains the two dimensional case was treated in d leykekhman and b vexler siam j numer anal 51 2013 pp 2797 2821 however the three dimensional case is technically much more involved to approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we establish mathcal o sqrt k h order of convergence rate for the control in the l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou siam j control optim 52 2014 pp 97 119 and in addition does not require any relationship between the time step k and the mesh size h the main technical tools are discrete maximal parabolic regularity results and the best approximation type estim

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

energy corrected finite element methods for corner singularities it is well known that the regularity of solutions of elliptic partial differential equations on domains with re entrant corners is limited by the maximal interior angle this results in reduced convergence rates for finite element approximations on families of quasi uniform meshes following an idea of zenger and gietl we show that it is possible to regain the full order of convergence by a local modification of the bilinear form in a vicinity of the singularity and thus to overcome the pollution effect a complete convergence analysis in weighted sobolev spaces is presented and we also show that the stress intensity factors can be computed with optimal accuracy the theoretical results are illustrated by numerical tests that demonstrate second order convergence of linear finite elements on families of quasi uniform meshes

homoclinic solutions of an infinite dimensional hamiltonian system we consider the system begin equation left begin array l partial t u delta x u v x u h v t x u v 3mm partial t v delta x v v x v h u t x u v end array qquad text for t x in mathbb r times mathbb r n right end equation which is an unbounded hamiltonian system in l 2 mathbb r n mathbb r 2m we assume that the constant function u 0 v 0 equiv 0 0 in mathbb r 2m is a stationary solution and that h and v are periodic in the t and x variables we present a variational formulation in order to obtain homoclinic solutions z u v satisfying z t x to 0 as t x to infty it is allowed that v changes sign and that delta v has essential spectrum below and above 0 we also treat the case of a bounded domain omega instead of mathbb r n with dirichlet boundary conditions

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

existence and multiplicity of homoclinic orbits for potentials on unbounded domains we study the system in r n where v is a potential with a strict local maximum at 0 and possibly with a singularity first using a minimising argument we can prove the existence of a homoclinic orbit when the component of x r n v x containing 0 is an arbitrary open set in the case unbounded we allow v x to go to 0 at infinity although at a slow enough rate then we show that the presence of a singularity in implies that a homoclinic solution can also be found via a minimax procedure and comparing the critical levels of the functional associated to the system we see that the two solutions are distinct whenever the singularity is not too far from 0

a posteriori error estimates for elliptic problems with dirac delta source terms the aim of this paper is to introduce residual type a posteriori error estimators for a poisson problem with a dirac delta source term in l p norm and w1 p seminorm the estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

genericity of the multibump dynamics for almost periodic duffing like systems in this paper we consider slowly oscillating perturbations of almost periodic du ng like systems i e systems of the form u u a t t w0 u t r u rn where w c2n rn r is superquadratic and a and are positive and almost periodic by variational methods we prove that if 0 is small enough then the system admits a multibump dynamics as a corollary we get that the system u u a t w0 u t r u rn admits multibump solutions whenever a a where a is an open dense subset of a c r r a is almost periodic and a t 0 t r

homoclinic orbits for second order self adjoint difference equations abstract in this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations p t u t 1 q t u t f t u t t z without any periodicity assumptions on p t q t and f providing that f t x grows superlinearly both at origin and at infinity or is an odd function with respect to x r and satisfies some additional assumptions

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

quasi optimal approximation of surface based lagrange multipliers in finite element methods we show quasi optimal a priori convergence results in the l 2 and h 1 2 norm for the approximation of surface based lagrange multipliers such as those employed in the mortar finite element method we improve on the estimates obtained in the standard saddle point theory where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estimates for the dual variable are reached for the lowest order case i e k 1 an additional factor of sqrt h ln h and for higher order cases i e k 1 an additional factor of sqrt h in the a priori bound for the dual variable can be recovered the proof is based on the use of new estimates for the primal variable in strips of width o h near these surfaces

existence of infinitely many homoclinic orbits for nonperiodic superquadratic hamiltonian systems abstract in this paper we study the following nonperiodic hamiltonian system z j h z t z where h t z is superquadratic in z r 2 n as z by applying a generalized linking theorem for strongly indefinite functionals we obtain infinitely many homoclinic orbits for the above system

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

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

homoclinic solutions of hamiltonian systems with symmetry abstract we prove the existence of infinitely many homoclinic solutions for a first order hamiltonian system symmetric with respect to an action of a compact lie group by means of variational methods we make no convexity assumption on the hamiltonian

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

infinitely many homoclinic orbits for the second order hamiltonian systems with general potentials abstract in this paper we study the following nonperiodic second order hamiltonian systems u t l t u t r t u t r where l t may not be uniformly positive definite for all t r under more general conditions on r u t u we prove that the above system has infinitely many homoclinic orbits provided r is even in u

homoclinics for asymptotically quadratic and superquadratic hamiltonian systems abstract in this paper we find new conditions which are different from those used in previous related studies to ensure the existence of infinitely many homoclinic orbits for the second order hamiltonian systems of the form q v q t q here we assume that v t q depends periodically on t and assume on q that v t q is asymptotically quadratic at q 0 and is as q either asymptotically quadratic or superquadratic as well as the new conditions

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a priori mesh grading for an elliptic problem with dirac right hand side the green function of the poisson equation in two dimensions is not contained in the sobolev space h 1 omega such that finite element error estimates for the discretization of a problem with the dirac measure on the right hand side are nonstandard and quasi uniform meshes are inappropriate by using graded meshes l 2 error estimates of almost optimal order are shown as a byproduct we show for the poisson equation with a right hand side in l 2 that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order

existence and exponential decay of homoclinics in a nonperiodic superquadratic hamiltonian system abstract this paper deals with existence and exponential decay of homoclinic orbits in the first order hamiltonian system z j h z t z where the hamiltonian function h t z is nonperiodic in t r and superquadratic in z r 2 n with certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

symmetrically homoclinic orbits for symmetric hamiltonian systems in this paper we study the existence of symmetric homoclinic orbits for first order and second order hamiltonian systems with some symmetric hamiltonian functions

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

homoclinic orbits for a nonperiodic hamiltonian system abstract in this paper we prove the existence and multiplicity of homoclinic orbits for first order hamiltonian systems of the form z j h z t z where h z is asymptotically linear at and is not assumed to be periodic

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

homoclinic orbits in a first order superquadratic hamiltonian system convergence of subharmonic orbits abstract we consider the existence of homoclinic orbits for a first order hamiltonian system z jh z t z we assume h t z is of form h t z 1 2 az z w t z where a is a symmetric matrix with ja i r and w t z is 2 periodic in t and has superquadratic growth in z we prove the existence of a nontrivial homoclinic solution z t and subharmonic solutions zt t t n i e 2 t periodic solutions of hs such that zt t z t in cloc1 r r 2n as t

existence and multiplicity of homoclinic orbits for the second order hamiltonian systems in this paper we study the following nonperiodic second order hamiltonian system u t l t u t ur t u t r where l t may not be uniformly positive definite for all t r and r t u is asymptotically quadratic in u as u by applying a generalized linking theorem in bartsch and ding 2006 8 we obtain the existence and multiplicity of homoclinic orbits for the above system

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

homoclinic orbits on compact manifolds abstract under a suitable assumption on the potential energy v we prove the existence of a homoclinic orbit of the equation dt x t grad v x t 0 x m where m is a compact riemannian manifold our assumption is satisfied for instance if the function v has a unique nondegenerate maximum point

finite element approximation of elliptic problems with dirac measure terms in weighted spaces applications to one and three dimensional coupled problems in this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving dirac measures using weighted sobolev spaces and weighted discrete norms our approach handles both the cases where the measure is simply a right hand side and where it represents an additional term i e solution dependent in the formulation of the problem the main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains where the coupling terms are seen as dirac measures concentrated on the fractures we first establish a decomposition lemma which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces then we consider the stability of the galerkin approximation with finite elements in weighted norms with uniform and graded meshes we introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf sup conditions in weighted norms then we focus on the convergence of the finite element method due to the lack of regularity the convergence rates are suboptimal for uniform meshes we show that using graded meshes optimal rates are recovered our theoretical results are supported by several numerical experiments finally we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three dimensional media with one dimensional fractures that are found in the analysis of microvascular flows

nontrivial solutions to a class of systems of second order differential equations abstract in this paper we investigate the nontrivial solutions to a class of systems of second order ordinary differential equations by using the critical groups and establish the existence results under the resonant case

connected finite loop spaces with maximal tori finite loop spaces are a generalization of compact lie groups however they do not enjoy all of the nice properties of compact lie groups for example having a maximal torus is a quite distinguished property actually an old conjecture due to wilkerson says said that every connected finite loop space with a maximal torus is equivalent to a compact connected lie group we give some more evidence for this conjecture by showing that the associated action of the weyl group on the maximal torus always represents the weyl group as a crystallographic group we also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected lie group if it has the right normalizer of the maximal torus actually in the cases under consideration the information about the weyl group is sufficient to give the answer all this is done by first studying the analoguous local problems