on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

existence and multiplicity of positive solutions for semilinear elliptic systems with sobolev critical exponents abstract in this paper we study the existence and multiplicity of positive solutions to the following system u f u u v e g x v f v u v e h x in u v 0 in and u v 0 on where is a bounded smooth domain in r n f c 1 r 2 r is positively homogeneous of degree g h c 1 0 and e is a positive parameter using sub supersolution method we prove the existence of positive solutions for the above problem by means of the variational approach we prove the multiplicity of positive solutions for the above problem with 2 2

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

preserving poisson structure and orthogonality in numerical integration of differential equations we consider the numerical integration of two types of systems of differential equations we first consider hamiltonian systems of differential equations with a poisson structure we show that symplectic runge kutta methods preserve this structure when the poisson tensor is constant using nonlinear changes of coordinates this structure can also be preserved for nonconstant poisson tensors as exemplified on the euler equations for the free rigid body we also consider orthogonal flows and the closely related class of isospectral flows to numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential algebraic equations daes and of integrating the system with a special combination of a particular class of runge kutta methods this approach requires only matrix matrix products and can preserve geometric properties of the flow such as reversibility

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

on constraint preservation in numerical simulations of yang mills equations the problem of constraint preservation for discretizations of nonlinear pdes is addressed in the example of the hyperbolic yang mills equations in temporal gauge these equations preserve a nonlinear divergence field analogous to the electric charge for maxwell s equations we introduce and discuss several discretizations of these equations on finite element spaces of lie algebra valued differential forms numerical experiments indicate that simply restricting the variational formulation to the galerkin spaces yields substantial drift in the charge contrary to the linear maxwell case we then propose a fully discrete method constrained with lagrange multipliers for which we prove discrete charge conservation and observe excellent energy conservation

a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e involving nonlinear boundary condition and sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

a lie poisson bracket formulation of plasticity and the computations based on the lie group so n abstract in this paper we develop a generalized hamiltonian formulation of a perfectly elastoplastic model which is a typical dissipative system on the cotangent bundle of the yield manifold a lie poisson bracket is used to construct the differential equations system the stress trajectory is a coadjoint orbit on the poisson manifold under a coadjoint action by the lie group so n the plastic differential equation is an affine non linear system of which a finite dimensional lie algebra can be constructed and the superposition principle is available for this system accordingly we can construct numerical schemes to automatically preserve the yield surface for perfect plasticity for isotropic hardening material as well as for an anisotropic elastic plastic model then we describe an anisotropic elastic plastic material model without entering the work hardening range and deforming under a specified dissipation rate which can be achieved through a stress dependent feedback control law of strain rate

a simplicial gauge theory we provide an action for gauge theories discretized on simplicial meshes inspired by lattice gauge theory and finite element methods the action is discretely gauge invariant and we give a proof of consistency a discrete noether s theorem that can be applied to our setting is also proved

multiple positive solutions for semi linear elliptic systems with sign changing weight abstract in this paper we study the multiplicity results of positive solutions for a semi linear elliptic system involving both concave convex and critical growth terms with the help of the nehari manifold and the lusternik schnirelmann category we investigate how the coefficient h x of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

existence of three nontrivial solutions for elliptic systems with critical exponents and weights abstract in this paper an elliptic system with critical sobolev exponents and weights is studied by the linking theorem on product space and the classical minimax theorem by investigating the coefficients of the critical nonlinearities we establish the existence of three nontrivial solutions

poisson integrators for lie poisson structures on mathbb r 3 this paper is concerned with the study of poisson integrators we are interested in lie poisson systems on first we focus on poisson integrators for constant poisson systems and the transformations used for transforming lie poisson structures to constant poisson structures then we construct local poisson integrators for lie poisson systems on finally we present the results of numerical experiments for two lie poisson systems and compare our poisson integrators with other known methods

multiple positive solutions for a critical quasilinear elliptic system with concave convex nonlinearities abstract in this paper we consider a quasilinear elliptic system with both concave convex nonlinearities and critical growth terms in bounded domains the existence and multiplicity results of positive solutions are obtained by variational methods

poisson schemes for hamiltonian systems on poisson manifolds abstract in this paper we construct poisson difference schemes of any order accuracy based on pade approximation for linear hamiltonian systems on poisson manifolds with constant coefficients for nonlinear hamiltonian systems on poisson manifolds we point out that symplectic diagonal implicit runge kutta methods are also poisson schemes the preservation of distinguished functions and quadratic first integrals of the original hamiltonian systems of these schemes are also discussed

multiple positive solutions of nonhomogeneous elliptic systems involving critical sobolev exponents abstract in this paper we study the existence of multiple positive solutions to the following elliptic system u 2 u u 2 v e f x v 2 u v v 2 e g x in u v 0 in and u v 0 on where is a bounded domain in r n n 3 f g c 1 and 1 for the subcritical and critical cases we prove that problem has at least two solutions for any e 0 e 0 and has no solution for any e e e 0 e in the supercritical case we find that the existence of solutions to problem is closely related to the existence of nonnegative solutions for two linear elliptic problems

the global existence of yang mills higgs fields in 4 dimensional minkowski space in this paper we complete the proof of global existence of yang mills higgs fields in 4 dimensional minkowski space by showing that an appropriate norm of the solutions cannot blow up in a finite time a key step in the proof is the demonstration that thel norm of the curvature is boundeda priori our results apply to any compact guage group and to any invariant higgs self coupling which is positive and of no higher than quartic degree

positive solutions for nonhomogeneous elliptic equations involving critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for the dirichlet problem u f z u q 2 u g z u 2 2 u in where 0 1 q 2 2 2 n n 2 0 r n n 3 is a bounded domain with smooth boundary and f g are nonnegative continuous functions on

local well posedness of the yang mills equation in the temporal gauge below the energy norm abstract we show that the yang mills equation in three dimensions in the temporal gauge is locally well posed in hs for s 3 4 if the hs norm is sufficiently small the temporal gauge is slightly less convenient technically than the more popular coulomb gauge but has the advantage of uniqueness even for large initial data and does not require solving a nonlinear elliptic problem to handle the temporal gauge correctly we project the connection into curl and divergence free components and develop some new bilinear estimates of xs b type which can handle integration in the time direction

constraint preserving schemes for some gauge invariant wave equations we study discretizations of the maxwell klein gordon equation as an example of a constrained geometric nonlinear evolution partial differential equation for the temporal gauge we propose a fully discrete scheme which preserves the nonlinear constraint thanks to a special application of lagrange multipliers we show that the method generalizes to hamiltonian wave equations whose kinetic and potential energy are both invariant under a group of transformations even though the galerkin spaces are not invariant we then extend the method to the lorenz gauge numerical results illustrate the discussion

simplicial gauge theory on spacetime we define a discrete gauge invariant yang mills higgs action on spacetime cylindrical meshes with simplicial spatial base the formulation is a generalization of classical lattice gauge theory and we prove consistency of the action in the finite element sense in addition we perform numerical tests of convergence towards exact continuum results for several choices of gauge fields in pure gauge theory

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

lie poisson integration for rigid body dynamics abstract in this paper the splitting midpoint rule is presented and proved to be the lie poisson integrators to the rigid body systems further discussions are also given numerical experiments show that this method has well properties comparing with the runge kutta method and ordinary midpoint rule

a div curl lemma for edge elements a variant of murat and tartar s div curl lemma is stated and proved for nedelec s edge elements given two sequences of vector fields of this type converging weakly in l2 as the mesh width tends to 0 we prove that their scalar product converges in the sense of distributions when one of the sequences consists of so called discrete divergence free fields whereas the other has relatively compact curl in h 1 the proof uses a uniform norm equivalence related to discrete compactness properties of vector finite element spaces and a super approximation property of scalar finite element spaces

multiplicity of positive solutions for a semilinear p laplacian system with sobolev critical exponent abstract in this paper we study the semilinear p laplacian system with critical growth terms in bounded domains by using the nehari manifold and variational methods we prove that the system has at least two positive solutions when the pair of the parameters belongs to a certain subset of r 2

a note for lie poisson hamilton jacobi equation and lie poisson integrator abstract in this paper a clear lie poisson hamilton jacobi theory is presented how to construct a lie poisson integrator by generating function methods is also given which is different from the ge marsden methods 1 an example on a rigid body has been given to illustrate this point

finite energy solutions of the yang mills equations in mathbb r 3 1 yang mills equations in r3 1 is well posed in the energy norm this means that for an appropriate gauge condition we construct local unique solutions in a time interval which depends only on the size of the energy norm of the data since the energy norm is left invariant by the yang mills flow the local solution is automatically extended to the entire space time thus our results which settle a problem stated in str imply the well known fundamental regularity result of eardley and moncrief e m that result proved in the temporal gauge requires a higher degree of smoothness for the data our main result proved also in the temporal gauge allows us to extend the concept of solutions to arbitrary finite energy initial data the solutions are automatically unique in the class of solutions obtained by our procedure moreover the global regularity proof given by eardley and moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat minkowski space time r3 1 namely the strong huygens principle indeed due mainly to this fact their proof does not seem to extend to general curved space times we have reasons to hope that the very different approach we take here will resolve this difficulty the basic ingredients of our method are 1 the introduction of appropriate local coulomb gauges adapted to the causal structure of the equations 2 an appropriate method of localizing the new space time estimates for

multiple positive solutions for a quasilinear elliptic system with critical exponent and sign changing weight abstract in this paper our main purpose is to establish the existence of multiple positive solutions of a quasilinear elliptic system with sobolev critical nonlinearities and sign changing weight by some standard variational methods by using the n ehari manifold fibering maps and the lusternik schnirelmann category we study the topology of the global maximum set f of f x and show the system has at least c a t f f 1 distinct positive solutions where cat f f denotes the lusternik schnirelmann category of f in f

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

multiple positive solutions for a critical elliptic system with concave convex nonlinearities we consider a semilinear elliptic system with both concave convex nonlinearities and critical growth terms in bounded domains the existence and multiplicity results of positive solutions are obtained by variational methods

multiple positive solutions for semilinear elliptic systems abstract this article investigates the effect of the coefficient f z of the critical nonlinearity for sufficiently small 0 there are at least k positive solutions of the semilinear elliptic systems u g z u p 2 u f z u 2 u v in v h z v p 2 v f z u v 2 v in u v 0 on where 0 r n is a bounded domain 1 1 and 2 p 2 for n 4

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells