effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

on the solution of stein s equation and fisher s information matrix of an armax process the main goal of this paper consists in expressing the solution of a stein equation in terms of the fisher information matrix fim of a scalar armax process a condition for expressing the fim in terms of a solution to a stein equation is also set forth such interconnections can be derived when a companion matrix with eigenvalues equal to the roots of an appropriate polynomial associated with the armax process is inserted in the stein equation the case of algebraic multiplicity greater than or equal to one is studied the fim and the corresponding solution to stein s equation are presented as solutions to systems of linear equations the interconnections are obtained by using the common particular solution of these systems the kernels of the structured coefficient matrices are described as well as some right inverses this enables us to find a solution to the newly obtained linear system of equations

an eigenvalue problem for hemivariational inequalities with combined nonlinearities on an infinite strip in this paper a class ofeigenvalue problems f hemivariational inequalities is studied which is defined on domains ofthe type r is a bounded open subset of r m m 1 and it involves

uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

fisher information in quantum statistics braunstein and caves braunstein s l and caves c m 1994 phys rev lett 72 3439 43 proposed to use helstrom s quantum information number to define meaningfully a metric on the set of all possible states of a given quantum system they showed that the quantum information is nothing other than the maximal fisher information in a measurement of the quantum system maximized over all possible measurements combining this fact with classical statistical results they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished based on arbitrary measurements on n identical copies of the given quantum system we show that the measurement which maximizes the fisher information typically depends on the true unknown state of the quantum system we close the resulting loophole in the argument by showing that one can still achieve the same optimal rate of distinguishability by a two stage adaptive measurement procedure when we consider states lying not on a smooth curve but on a manifold of higher dimension the situation becomes much more complex we show that the notion of distinguishability of close by states depends strongly on the measurement resources one allows oneself and on a further specification of the task at hand the quantum information matrix no longer seems to play a central role

semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

hylomorphic solitons for the benjamin ono and the fractional kdv equations abstract this paper concerns with the existence of solitons namely stable solitary waves for the benjamin ono and the fractional kdv equations

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

the bezoutian state space realizations and fisher s information matrix of an arma process in this paper we derive some properties of the bezout matrix and relate the fisher information matrix for a stationary arma process to the bezoutian some properties are explained via realizations in state space form of the derivatives of the white noise process with respect to the parameters a factorization of the fisher information matrix as a product in factors which involve the bezout matrix of the associated ar and ma polynomials is derived from this factorization we can characterize singularity of the fisher information matrix

upper bound of weak limitation for the gross pitaevskii equation abstract this paper is concerned with the blow up solutions of gross pitaevskii equation we obtain the upper bound of weak limitation for the blow up solutions by using the method of cazenave 2003 3 as well as the concentration compact principle

total variation diminishing implicit explicit runge kutta methods for the simulation of double diffusive convection in astrophysics we put forward the use of total variation diminishing or more generally strong stability preserving implicit explicit runge kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection the fully compressible navier stokes equation augmented by continuity and total energy equations and an equation of state describing the relation between the thermodynamic quantities is semi discretized in space by essentially non oscillatory schemes and dissipative finite difference methods it is subsequently integrated in time by runge kutta methods which are constructed such as to preserve the total variation diminishing or strong stability property satisfied by the spatial discretization coupled with the forward euler method we analyse the stability accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit runge kutta methods

existence of multiple positive solutions of subcritical semilinear elliptic problems in exterior strip domains abstract in this paper we consider the subcritical semilinear elliptic problem u u k x u p f x in u 0 in u h 0 1 where 0 n 3 1 p n 2 n 2 and is an exterior strip domain in r n under some suitable conditions on k and f we show that there exists a positive constant depending on k and f such that has exactly two solutions if 0 and no solution if furthermore if there exists a positive constant k 0 such that k x k 0 for x and f is bounded in then has at least one solution for

on the lowest eigenvalue of general operators for the intersection of two domains e lieb invent math 74 1983 441 448 has proved that ifaandbare two bounded domains in rn then there exists a translation ysuch that 1 a yb 1 a 1 b where 1is the first eigenvalue of the laplacian with dirichlet boundary conditions here we extend this result to elliptic operators in divergence froml div q x c x with mixed dirichlet neumann boundary conditions ona if l a is the corresponding eigenvalue we show that there exists a translation ysuch that l a yb l a 1 b ift q x 2for allx a rn one can further improve the estimate for non isotropic operators where can be large by taking into account rotations ofb in that case a similar inequality holds ift q x 2 whereqis a mean value ofqin different directions

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

strong stability for additive runge kutta methods space discretization of some time dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties in these situ

a non homogeneous elliptic problem in low dimensions with three symmetric solutions abstract we investigate the existence of multiple solutions for a low dimensional neumann problem with non standard growth set on a strip like domain of r n variational techniques depending in particular on a minimax theorem and a version of the principle of symmetric criticality are employed to establish the existence of three non zero solutions endowed with axial symmetry some consequences are also derived

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

an existence result for gradient type systems with a nondifferentiable term on unbounded strips in this paper we study the existence of nontrivial solutions for a class of gradient type systems on strip like domains where the nonlinear term is not necessarily continuously differentiable the proof of the main result is based on a nonsmooth version of the mountain pass theorem which involves the cerami compactness condition and on the principle of symmetric criticality for locally lipschitz functions

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

two classes of implicit explicit multistep methods for nonlinear stiff initial value problems the initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications in order to reduce the computation cost implicit explicit imex methods are often applied to these problems i e the stiff and non stiff terms are discretized by using implicit and explicit methods respectively in this paper we mainly consider the nonlinear stiff initial value problems satisfying the one sided lipschitz condition and a class of singularly perturbed initial value problems and present two classes of the imex multistep methods by combining implicit one leg methods with explicit linear multistep methods and explicit one leg methods respectively the order conditions and the convergence results of these methods are obtained some efficient methods are constructed some numerical examples are given to verify the validity of the obtained theoretical results

decay of heat semigroups in l and applications to nonlinear parabolic problems in unbounded domains we characterize all domains of rn such that the heat semigroup decays in l l or l l1 as t namely we prove that this property is equivalent to the poincare inequality and that it is also equivalent to the solvability of u f in l for all f l in particular under mild regularity assumptions on these properties are equivalent to the geometric condition that has finite inradius next we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains first we consider the quenching problem for singular parabolic equations we prove that the solution in quenches in finite time no matter how small the nonlinearity is if and only if does not fulfill the poincare inequality second for the semilinear heat equation with a power nonlinearity we prove roughly speaking that the trivial solution is stable in l or in l1 if and only if has finite inradius

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

a variational inequality on the half line abstract multiple solutions are obtained for a variational inequality defined on the half line 0 our approach is based on a key embedding result as well as on the non smooth critical point theory for szulkin type functionals

large heterocline orbits for some evolution equations soit s r ou est un domaine borne de r n avec une frontiere reguliere on etudie l existence des orbites heteroclines pour u f u 0 dans s u 0 sur s avec f c 1 r r satisfaisant certaines conditions

the focusing singularity of the davey stewartson equations for gravity capillary surface waves abstract we analyze the self focusing of gravity capillary surface waves modeled by the davey stewartson equations this system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two dimensional schrodinger equation with a non local cubic nonlinearity with accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical as in the usual two dimensional cubic schrodinger equation which is the deep water limit of the davey stewartson equations we also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

an explicit expression for the fisher information matrix of a multiple time series process the principal result in this paper is concerned with the derivative of a vector with respect to a block vector or matrix this is applied to the asymptotic fisher information matrix fim of a stationary vector autoregressive and moving average time series process varma representations which can be used for computing the components of the fim are then obtained in a related paper a klein a generalization of whittle s formula for the information matrix of vector mixed time series linear algebra appl 321 2000 197 208 the derivative is taken with respect to a vector this is obtained by vectorizing the appropriate matrix products whereas in this paper the corresponding matrix products are left unchanged

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

multirate generalized additive runge kutta methods this work constructs a new class of multirate schemes based on the recently developed generalized additive runge kutta gark methods sandu and gunther siam j numer anal 53 1 17 42 2015 multirate schemes use different step sizes for different components and for different partitions of the right hand side based on the local activity levels we show that the new multirate gark family includes many well known multirate schemes as special cases the order conditions theory follows directly from the gark accuracy theory nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold

travelling wave solutions for the discrete sine gordon equation with nonlinear pair interaction the focus of study is the nonlinear discrete sine gordon equation where the nonlinearity refers to a nonlinear interaction of neighbouring atoms the existence of travelling heteroclinic homoclinic and periodic waves is shown the asymptotic states are chosen such that the action functional is finite the proofs employ variational methods in particular a suitable concentration compactness lemma combined with direct minimisation and mountain pass arguments

a nonsmooth principle of symmetric criticality and variational hemivariational inequalities in this paper we prove the principle of symmetric criticality for motreanu panagiotopoulos type functionals i e for convex proper lower semicontinuous functionals which are perturbed by a locally lipschitz function by means of this principle a variational hemivariational inequality is studied on certain type of unbounded strips

on the asymptotic properties of imex runge kutta schemes for hyperbolic balance laws implicit explicit imex schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources here we focus our attention on the asymptotic properties of such schemes like the preservation of steady states well balanced property and the behavior in presence of small space time scales asymptotic preservation property we analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well balanced behavior in addition we consider a partitioned strategy which possesses better well balanced properties the behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic preserving schemes additional order conditions that guarantee the correct behavior of the schemes in the navier stokes regime are derived several examples illustrate these asymptotic behaviors and the performance of the new methods

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

harnack s inequality for a nonlinear eigenvalue problem on metric spaces abstract we prove harnack s inequality for first eigenfunctions of the p laplacian in metric measure spaces the proof is based on the famous moser iteration method which has the advantage that it only requires a weak 1 p poincare inequality as a by product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains

on the variational approach to semilinear elliptic equations with scale covariance abstract semilinear elliptic equations on r d d 3 with some scale covariance are considered in the variational approach at first it is explained when a minimizer of the associated constrained minimization problem gives rise to a solution of the differential equation then the existence problem for a minimizer is solved on an abstract level the importance of the symmetrygroup of the constrained minimization problem is stressed finally some examples are treated explicitly a new proof for some previous results is given and a comparison with other recent results is made

error threshold in simple landscapes we consider the quasispecies description of a population evolving in both the master sequence landscape where a single sequence is evolutionarily preferred over all others and the rem landscape where the fitness of different sequences is an independent identically distributed random variable we show that in both cases the error threshold is analogous to a first order thermodynamical transition where the overlap between the average genotype and the optimal one drops discontinuously to zero

on the sign of the mountain pass solution we consider a class of superlinear elliptic problems on a bounded domain in r n i t is known that besides the zero solution such superlinear problems possess at least one positive and one negative solution of mountain pass type as well as a sign changing solution moreover the least energy solutions ground states are among those one sign solutions we study the eect of the nonlinearity on the sign of such ground states and the corresponding role played by the interior geometry of and the geometry of 2001 elsevier science ltd all rights reserved

cauchy problem for the l2 critical nonlinear schrodinger equation below h1 abstract in this paper we consider solutions of the l 2 critical focusing nonlinear schrodinger equation on r 2 for the initial data in h s s 1 mainly about the l 2 concentration phenomena for blow up solutions and global well posedness for small initial data

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

a generalized structure approach to additive runge kutta methods this work considers a general structure of the additively partitioned runge kutta methods by allowing for different stage values as arguments of different components of the right hand side an order conditions theory is developed for the new formulation of generalized additive methods and stability and monotonicity investigations are carried out the paper discusses the construction and properties of implicit explicit and implicit implicit methods in the new framework the new approach named gark introduces additional flexibility when compared to traditional partitioned runge kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales or driven by multiple physical processes

faber krahn type inequalities and uniqueness of positive solutions on metric measure spaces abstract we consider a general class of metric measure spaces equipped with a strongly local regular dirichlet form and provide a lower bound on the hitting time probabilities of the associated hunt process using these estimates we establish i a generalization of the classical lieb s inequality and ii uniqueness of nonnegative super solutions to semi linear elliptic equations on metric measure spaces finally using heat kernel estimates we generalize the local faber krahn inequality recently obtained in 28 to local and non local dirichlet spaces

existence and multiplicity results for hemivariational inequalities with two parameters a multiplicity result of ricceri based on a minimax inequality is applied to locally lipschitz functionals and the existence of one or two non trivial solutions for a class of two parameter hemivariational inequalities is proved an application is given to elliptic problems with discontinuous nonlinearities on unbounded domains

the uniqueness of hill s spherical vortex the mathematical description of steady vortex rings in an ideal fluid occupying the whole space r3 can be approached in various ways the physical basis of the problem its history up to 1973 and several formulations are outlined in 12 pp 14 21 another quite different formulation and the plan for a corresponding existence theory are presented in 6 further existence theorems variational principles and results are to be found in 3 7 11 13 and 20 here we state only definitions and equations that seem relevant to our immediate purpose

a variational approach to the isoperimetric inequality for the robin eigenvalue problem the isoperimetric inequality for the first eigenvalue of the laplace operator with robin boundary conditions was recently proved by daners in the context of lipschitz sets this paper introduces a new approach to the isoperimetric inequality based on the theory of special functions of bounded variation sbv we extend the notion of the first eigenvalue 1 for general domains with finite volume possibly unbounded and with irregular boundary and we prove that the balls are the unique minimizers of 1 among domains with prescribed volume

minimum action solutions of non linear elliptic equations in unbounded domains containing a plane let d 1 be an integer and d a smooth bounded domain and consider the elliptic equation u g u on 2 we prove that under almost necessary and sufficient conditions on the continuous function g m m the above equation has a minimum action solution

a mountain pass algorithm with projector in this paper we present an enhanced version of the minimax algorithm of chen ni and zhou that offers the additional guarantee that the solution found is a fix point of a projector on a cone positivity negativity and monotonicity can be expressed in this way the convergence of the algorithm is proved by means of a computational deformation lemma instead of the usual deformation lemma used in the calculus of variations

on the asymptotics of nodes ofl2 solutions of schrodinger equations in dimensions 3 let r n br wheren 3 andbr x n x r we investigate the asymptotics of real valued solutions l2 r of the schrodinger equation v e 0 wheree 0 begin gathered mathop lim inf limits r to infty r gamma int limits s r psi 2 d sigma int limits s r psi 2 d sigma 0 and hfill mathop lim inf limits r to infty ln volume d cap b r ln r geqq n 1 2 hfill end gathered results of this type are already non trivial for radial problems with satisfying non radial boundary conditions on r or for excited states of the hydrogen atom if one considers linear combinations of differentl waves

on stability issues for imex schemes applied to 1d scalar hyperbolic equations with stiff reaction terms the application of a method of lines to a hyperbolic pde with source terms gives rise to a system of odes containing terms that may have very different stiffness properties in this case implicit explicit runge kutta imex rk schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms which can be discretized using the highly developed shock capturing technology together with an implicit treatment of the source terms necessary for stability reasons motivated by the structure of the source term in a model problem introduced by leveque and yee in j comput phys 86 1990 in this paper we study the preservation of certain invariant regions as a weak stability constraint for the class of source terms considered in this paper the unit interval is an invariant region for the model balance law in the first part of the paper we consider first order time discretizations which are the basic building blocks of higher order imex rk schemes and study the conditions that guarantee that 0 1 is also an invariant region for the numerical scheme in the second part of the paper we study the conditions that ensure the preservation of this property for higher order imex schemes

minimization problems in l1 r3 abstract in this paper minimization problems in l1 ir3 are considered these problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids rotating stars under nearly optimal assumptions a minimizer is proved to exist by a direct variational method which uses heavily the symmetry of the problem in order to get some compactness finally by looking directly at the euler equation we give some existence results of solutions of the euler equation even if the infimum is not finite author

on fisher s information matrix of an armax process and sylvester s resultant matrices we establish a relation between fisher s information matrix of a stationary autoregressive moving average process with an exogenous component and two sylvester s resultant matrices

existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends abstract in this paper we study the existence of positive solutions to nonlinear elliptic boundary value problems on unbounded domains r n with cylindrical ends for a general nonlinear term f u including f u u p 1 n 2 as a typical example u u f u u 0 x u 0 u x 0 x by using the mountain pass approach the geometry of plays an important role in our analysis

traveling waves for nonlinear schrodinger equations with nonzero conditions at infinity for a large class of nonlinear schrodinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension n 3 our results are valid as well for the gross pitaevskii equation and for nls with cubic quintic nonlinearity

maximizers for gagliardo nirenberg inequalities and related non local problems in this paper we study the existence of maximizers for two families of interpolation inequalities namely a generalized gagliardo nirenberg inequality and a new inequality involving the riesz energy two basic tools in our argument are a generalization of lieb s translation lemma and a riesz energy version of the brezis lieb lemma

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

boundedness and strong stability of runge kutta methods in the literature much attention has been paid to runge kutta methods rkms satisfying special nonlinear stability requirements indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many rkms the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we study stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we focus on stepsize conditions which are optimal in that they are not unnecessarily restrictive we nd that in situations where the special stability properties mentioned above are violated boundedness can be present only within a class of very special rkms as a byproduct our analysis sheds new light on the known theory of monotonicity for rkms we obtain separate results for internal and external monotonicity as well as a new proof of the fundamental relation between monotonicity and kraaijevanger s coecient this proof distinguishes itself from older ones in that it is shorter and more transparent while it requires more simple assumptions on the rkms under consideration

a block hankel generalized confluent vandermonde matrix abstract vandermonde matrices are well known they have a number of interesting properties and play a role in lagrange interpolation problems partial fraction expansions and finding solutions to linear ordinary differential equations to mention just a few applications usually one takes these matrices square q q say in which case the i th column is given by u z i where we write u z 1 z z q 1 if all the z i i 1 q are different the vandermonde matrix is non singular otherwise not the latter case obviously takes place when all z i are the same z say in which case one could speak of a confluent vandermonde matrix non singularity is obtained if one considers the matrix v z whose i th column i 1 q is given by the i 1 th derivative u i 1 z we will consider generalizations of the confluent vandermonde matrix v z by considering matrices obtained by using as building blocks the matrices m z u z w z with u z as above and w z 1 z z r 1 together with its derivatives m k z specifically we will look at matrices whose ij th block is given by m i j z where the indices i j by convention have initial value zero these in general non square matrices exhibit a block hankel structure we will answer a number of elementary questions for this matrix what is the rank what is the null space can the latter be parametrized in a simple way does it depend on z what are left or right inverses it turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure the answers depend on the size of the matrix m z and the number of derivatives m k z that is involved the results are obtained by mostly elementary methods no specific knowledge of the theory of matrix polynomials is needed

on the resultant property of the fisher information matrix of a vector arma process a matrix is called a multiple resultant matrix associated to two matrix polynomials when it becomes singular if and only if the two matrix polynomials have at least one common eigenvalue in this paper a new multiple resultant matrix is introduced it concerns the fisher information matrix fim of a stationary vector autoregressive and moving average time series process varma the two matrix polynomials are the autoregressive and the moving average matrix polynomials of the varma process in order to show that the fim is a multiple resultant matrix two new representations of the fim are derived to construct such representations appropriate matrix differential rules are applied the newly obtained representations are expressed in terms of the multiple sylvester matrix and the tensor sylvester matrix the representation of the fim expressed by the tensor sylvester matrix is used to prove that the fim becomes singular if and only if the autoregressive and moving average matrix polynomials have at least one common eigenvalue it then follows that the fim and the tensor sylvester matrix have equivalent singularity conditions in a simple numerical example it is shown however that the fim fails to detect common eigenvalues due to some kind of numerical instability whereas the tensor sylvester matrix reveals it clearly proving the usefulness of the results derived in this paper

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

geometric methods for nonlinear many body quantum systems abstract geometric techniques have played an important role in the seventies for the study of the spectrum of many body schrodinger operators in this paper we provide a formalism which also allows to study nonlinear systems we start by defining a weak topology on many body states which appropriately describes the physical behavior of the system in the case of lack of compactness that is when some particles are lost at infinity we provide several important properties of this topology and use them to write a simple proof of the famous hvz theorem in the repulsive case in the second step we recall the method of geometric localization in fock space as proposed by derezinski and gerard and we relate this tool to our weak topology we then provide several applications we start by studying the so called finite rank approximation which consists in imposing that the many body wavefunction can be expanded using finitely many one body functions we thereby emphasize geometric properties of hartree fock states and prove nonlinear versions of the hvz theorem in the spirit of works of friesecke in the last section we study translation invariant many body systems comprising a nonlinear term which effectively describes the interactions with a second system as an example we prove the existence of the multi polaron in the pekar tomasevich approximation for certain values of the coupling constant

a hemivariational inequality involving p laplacian on an infinite strip abstract this paper deals with an eigenvalue problem for hemivariational inequalities on domains of the type r is a bounded open subset of r n 1 n 2 and it involves concave convex nonlinearities under suitable conditions on the nonlinearities two nontrivial solutions are obtained which belong to a special closed convex cone of w 0 1 p r whenever the eigenvalues are of certain range our approach is variational based on the theories of non smooth analysis our results are a generalization of the case of laplacian from a kristaly et al to the case of p laplacian