free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

nonlinear hemivariational inequalities at resonance in this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue 1 of h01 z we prove two existence theorems for such problems our approach is variational and is based on the nonsmooth critical point theory of chang which uses the subdifferential calculus of clarke for locally lipschitz functions

periodic solutions for asymptotically linear hamiltonian system with resonance both at infinity and origin via computation of critical groups abstract in this paper we study the existence of nontrivial periodic solutions for asymptotically linear hamiltonian system which has the same linear term both at infinity and origin by computation of critical groups at origin and infinity associated with maslov type index we establish the existence for nontrivial periodic solution of the system moreover some conditions employed for the computation of critical groups are weakened

multiple positive solutions for nonlinear periodic problems abstract we consider a nonlinear periodic equation driven by the scalar p laplacian and with a caratheodory asymptotically p 1 linear nonlinearity using variational methods coupled with suitable truncation techniques we show that the problem has at least two positive solutions for the semilinear case p 2 using morse theory we show that the problem has three distinct positive solutions

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

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

undecidability of automorphism groups we give a simple and easy to apply technique that gives the undecidability of the theory of many automorphism groups let g be a group of automorphisms of a structure suppose that g in g is not the identity and has no non singleton finite orbits if the centraliser of g is transitive on the support of g and satisfies a further technical condition then the subgroup generated by g is equal to the double centraliser of g thus if g contains such an element g that is conjugate to all its positive powers then one can interpret addition and multiplication of natural numbers in the theory of g using the parameter g consequently g has undecidable theory

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

numerical simulation of spray coalescence in an eulerian framework direct quadrature method of moments and multi fluid method the scope of the present study is eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation the fundamental mathematical description is the williams spray equation governing the joint number density function f v u x t of droplet volume and velocity eulerian multi fluid models have already been rigorously derived from this equation in laurent et al f laurent m massot p villedieu eulerian multi fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays j comput phys 194 2004 505 543 the first key feature of the paper is the application of direct quadrature method of moments dqmom introduced by marchisio and fox d l marchisio r o fox solution of population balance equations using the direct quadrature method of moments j aerosol sci 36 2005 43 73 to the williams spray equation both the multi fluid method and dqmom yield systems of eulerian conservation equations with complicated interaction terms representing coalescence in order to focus on the difficulties associated with treating size dependent coalescence and to avoid numerical uncertainty issues associated with two way coupling only one way coupling between the droplets and a given gas velocity field is considered in order to validate and compare these approaches the chosen configuration is a self similar 2d axisymmetrical decelerating nozzle with sprays having various size distributions ranging from smooth ones up to dirac delta functions the second key feature of the paper is a thorough comparison of the two approaches for various test cases to a reference solution obtained through a classical stochastic lagrangian solver both eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the lagrangian solver the third key point of the study is a detailed description of the limitations associated with each method thus giving criteria for their use as well as for their respective efficiency

nonlinear degenerate elliptic equation with hardy potential and critical parameter abstract some embedding inequalities in hardy sobolev spaces with general weight functions were proved and a positive answer to an open problem raised by brezis vazquez was given in the weighted hardy sobolev spaces the existence of nontrivial many solutions to the corresponding nonlinear degenerated elliptic equations with hardy potential and critical parameter under conditions weaker than ambrosetti rabinowitz condition was obtained

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where 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 our assumptions on k and f are rather loose

conditional quadrature method of moments for kinetic equations kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution moment methods are an important class of approximate models derived from kinetic equations but require closure to truncate the moment set in quadrature based moment methods qbmm closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments e g spatial fluxes can be related to the finite moment set in this work a novel moment inversion algorithm based on 1 d adaptive quadrature of conditional velocity moments is introduced and shown to always yield realizable distribution functions i e non negative quadrature weights this conditional quadrature method of moments cqmom can be used to compute exact n point quadratures for multi valued solutions also known as the multi variate truncated moment problem and provides optimal approximations of continuous distributions in order to control numerical errors arising in volume averaging and spatial transport an adaptive 1 d quadrature algorithm is formulated for use with cqmom the use of adaptive cqmom in the context of qbmm for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing i e collision less systems elastic and inelastic particle particle collisions and external forces i e fluid drag

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear hamiltonian system in this paper we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear hamiltonian systems with impulsive conditions via variational methods however without impulses there is no homoclinic or periodic solution for the system considered in this paper moreover our results can be used to study the existence of periodic and homoclinic solutions of difference equations

notes on periodic solutions of subquadratic second order systems some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory

periodic solutions to second order hamiltonian systems in an unbounded potential well in this paper we give some results on the existence of periodic solutions to the second order hamiltonian system where and is an open set of n with non empty bounded complement n we suppose v t x is periodic in t v t x as x and v is super or sub quadratic as x

remarks on a brezis nirenberg s result in this note we revisit the equation with sobolev critical exponent which is firstly studied by brezis nirenberg in 4 p u u u p 2u u 2 2u in u h01 where rn is a bounded smooth domain n 3 2 0 r we show that for any m n there exists m 0 such that p has m solutions for any r 0 and every 0 m when n 4 0 we show that there is 0 0 such that problem p has a nontrivial solution for every 0 moreover we get multiple sign changing solutions for problem p

an asymptotically linear non cooperative elliptic system with lack of compactness in this paper we consider an asymptotically linear non cooperative elliptic system at resonance the functional associated to the system is strongly indefinite and so the morse index of each critical point is infinite to overcome this difficulty we will use morse theory for strongly indefinite functionals developed by abbondandolo we shall prove that if the system has a non resonant solution the morse index of which is different from the morse index at infinity then there exists another solution

a level set approach for dilute non collisional fluid particle flows gas particle and other dispersed phase flows can be described by a kinetic equation containing terms for spatial transport acceleration and particle processes such as evaporation or collisions however computing the dispersed velocity is a challenging task due to the large number of independent variables a level set approach for computing dilute non collisional fluid particle flows is presented we will consider the sprays governed by the williams kinetic equation subject to initial distributions away from equilibrium of the form i 1 n i x u i x the dispersed velocity is described as the zero level set of a smooth function which satisfies a transport equation this together with the density weight recovers the particle distribution at any time moments of any desired order can be evaluated by a quadrature formula involving the level set function and the density weight it is shown that the method can successfully handle highly non equilibrium flows e g impinging particle jets jet crossing particle rebound off walls finite stokes number flows

simulation of fluid and particles flows asymptotic preserving schemes for bubbling and flowing regimes in this work we propose asymptotic preserving numerical schemes for the bubbling and flowing regimes of particles immersed in a fluid treated by two phase flow models the description comprises compressible euler equations for the dense phase fluid and a kinetic fokker planck equation for the disperse phase particles coupled through friction terms we show numerical simulations in the relevant case of gravity in the one dimensional case demonstrating the overall behavior of the schemes