liouville theorem for elliptic equations with nonlinear boundary value conditions and finite morse indices abstract in this paper we study the nonexistence result for the following nonlinear elliptic equation with nonlinear boundary value condition u u p 1 u in r n u u q 1 u on r n if 1 p n 2 n 2 1 q n n 2 and p q n 2 n 2 n n 2 then we will show that this problem does not possess nontrivial bounded solution with finite morse index

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

stable and finite morse index solutions on ⁿ or on bounded domains with small diffusion in this paper we study bounded solutions of u f u on r n where n 2 and sometimes n 3 and show that for most f s the weakly stable and finite morse index solutions are quite simple we then use this to obtain a very good understanding of the stable and bounded morse index solutions of e 2 u f u on with dirichlet or neumann boundary conditions for small e

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

estimate of morse index of cooperative elliptic systems and its application to spatial vector solitons local instability index of unstable solutions to single partial differential equations pdes by a local minimax method lmmm was established in zhou 2005 it is known that the local min orthogonal method lmom which was first proposed in zhou 2004 and then further developed in chen et al 2008 can find more general unstable solutions to both single pdes and cooperative elliptic systems this paper is to carry out instability analysis of unstable solutions by lmom to which an infinite dimensional functional space can be decomposed as a direct sum of a finite dimensional subspace and its orthogonal complement a morse index approach is developed to show that with lmom instability behavior of a solution in such infinite dimensional complement subspace can be totally determined usual instability analysis in an entire space is then reduced to analysis in its finite dimensional subspace for which a corresponding matrix decomposition is proposed to analyze a solution s instability behavior estimates of morse index are also established finally numerical examples of both 2 and 3 component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons whose local instabilities are numerically confirmed by the new estimates certain important properties of the examples are also verified or presented

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

multiple solutions for semilinear elliptic systems with non homogeneous boundary conditions since the early seventies many authors have widely investigated existence and multiplicity of solutions for semilinear elliptic problems with dirichlet boundary conditions especially by means of variational methods see 22 and references therein in particular if is a real l2 function on a bounded domain rn p 2 and pi 2 if n 3 here 2 2n n 2 the following model problem 6u u p 2u in

four block problem stable plants and rational weights it is shown that for a large class of multivariable infinite dimensional stable plants and rational weights the four block problem reduces to finding the singular values of a certain operator defined by feintuch and francis 1986 via the singular system which is studied by foias and tannenbaum 1988 and ozbay and tannenbaum 1988

a conservative semi lagrangian multi tracer transport scheme cslam on the cubed sphere grid a conservative multi tracer transport algorithm on the cubed sphere based on the semi lagrangian approach cslam has been developed the scheme relies on backward trajectories and the resulting upstream cells polygons are approximated with great circle arcs biquadratic polynomial functions are used for approximating the density distribution in the cubed sphere grid cells the upstream surface integrals associated with the conservative semi lagrangian scheme are computed as line integrals by employing the gauss green theorem the line integrals are evaluated using a combination of exact integrals and high order gaussian quadrature the upstream cell trajectories information and computation of weights of integrals can be reused for each additional tracer the cslam scheme is extensively tested with various standard benchmark test cases of solid body rotation and deformational flow in both cartesian and spherical geometry and the results are compared with those of other published schemes the cslam scheme is accurate robust and moreover the edges and vertices of the cubed sphere discontinuities do not affect the overall accuracy of the scheme the cslam scheme exhibits excellent convergence properties and has an option for enforcing monotonicity the advantages of introducing cross terms in the fully two dimensional biquadratic density distribution functions are also examined in the context of cartesian as well as the cubed sphere grid which has six local sub domains with discontinuous edges and corners

numerical wave propagation on non uniform one dimensional staggered grids the wave propagation behaviour of centered difference schemes on one dimensional non uniform staggered grids is investigated previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids for waves of a given frequency the wave field is decomposed into right and left propagating components and a wave energy conservation law is derived in terms of these components for slowly varying grids separate evolution equations for the right and left propagating components are derived leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid provided that waves of that frequency are resolvable however there will be reflection from any location at which the wave group velocity goes to zero the possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted grids with an abrupt jump in resolution are also investigated it is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump however it does not appear possible to avoid complete reflection except by introducing extra dissipation terms if the waves are not resolvable on one side of the jump an example is presented of a second order accurate scheme that spontaneously radiates waves from the resolution jump

adaptive grid refinement for numerical weather prediction abstract an adaptive atmospheric flow model is described and results of integrations with this model are presented the adaptive technique employed is that of berger and oliger the technique uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids which have been placed based on a richardson type estimate of the truncation error in the coarse grid solution by correctly coupling the integrations on the various grids periodically re estimating the error and recreating the finer grids uniformily accurate solutions are economically produced the primitive hydrostatic equations of meteorology are solved for the advection of a barotropic cyclone and for the development of a baroclinic disturbance which results from the perturbation of an unstable jet these integrations demonstrate the feasibility of using multiple rotated overlapping fine grids direct computations of the truncation error are used to confirm the accuracy of the richardson type truncation error estimates

on a nonlinear elliptic eigenvalue problem abstract let n be the number of the solutions of the equation u f u in u 0 on where is a bounded domain in r n n 2 with smooth boundary under suitable conditions on f we proved that n as

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

tutorial review h optimal controller design for a class of distributed parameter systems this paper is a tutorial on a frequency domain design method for the two block h optimal control of a class of siso distributed parameter systems uncertainties in the system are assumed to be unmodelled dynamics represented in the form of either multiplicative or additive or coprime factor perturbations of a nominal infinite dimensional plant performance of the closed loop system is measured in terms of the energy amplification from external input signals to the outputs of interest in this paper transfer functions of the nominal plant and the weights are used in order to derive a solution to two block h control problems the solution presented here uses results from operator theory the h optimal performance and controller in this method are computed from a finite set of linear equations

morse indices of solutions for super linear elliptic pde s abstract in this paper we establish l estimates for solutions of some general superlinear and subcritical elliptic equations via the morse index our results generalize and improve the previous works bahri and lions 1992 yang 1998

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

an approximation approach to h control problems for distributed parameter systems a new approximation approach is presented for h control problems involving distributed parameter systems related approximation error bound is provided using frostman s theorem we approximate a general inner function with a finite blaschke product and then apply known finite dimensional solution techniques this simple method can give an approximate solution of various infinite dimensional h control problems with great generality

an analysis of 1d finite volume methods for geophysical problems on refined grids this paper examines high order unstaggered symmetric and upwind finite volume discretizations of the advection equation in the presence of an abrupt discontinuity in grid resolution an approach for characterizing the initial amplitude of a parasitic mode as well as its decay rate away from a grid resolution discontinuity is presented using a combination of numerical analysis and empirical studies it is shown that spurious parasitic modes which are artificially generated by the resolution discontinuity are mostly undamped by symmetric finite volume schemes but are quickly removed by upwind and semi lagrangian integrated mass slim schemes slope curvature limiting is insufficient to completely remove these modes especially at low forcing frequencies where the incident wave can act as a carrier of the parasitic mode increasing the order of accuracy of the reconstruction at the grid interface is effective at removing noise from the lowest frequency incident modes but insufficient at high frequencies it is shown that this analysis can be extended to the 1d linear shallow water equations via riemann invariants

existence of radial solutions with prescribed number of zeros for elliptic equations and their morse index abstract in this paper we consider radially symmetric solutions of the nonlinear dirichlet problem u f x u 0 in where is a ball in r n n 3 and f satisfies some appropriate assumptions we prove existence of radially symmetric solutions with k prescribed number of zeros moreover when f x u k x u p 1 u using the uniqueness result due to tanaka 2008 21 we verify that these solutions are non degenerate and we prove that their radial morse index is exactly k

exact multiplicity of solutions to superlinear and sublinear problems f2 lim u f u u f 0 lim u f u u f 0 for the de5niteness we assume f f and when f f we use f to represent it we will consider f being either superlinear or sublinear f is said to be superlinear if f u u is decreasing in 0 and is increasing in 0 and f is said to be sublinear if f u u is increasing in 0 and is decreasing in 0 the semilinear equation 1 1 with f satisfying f1 f2 has been studied extensively since early 1970s several di9erent approaches like variational methods morse

numerical representation of geostrophic modes on arbitrarily structured c grids a c grid staggering in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces or edges in two dimensions is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes however the discretization of the coriolis terms is non trivial for constant coriolis parameter the linearized shallow water equations support geostrophic modes stationary solutions in geostrophic balance a naive discretization of the coriolis terms can cause geostrophic modes to become non stationary causing unphysical behaviour of numerical solutions recent work has shown how to discretize the coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the coriolis terms remain energy conserving in this paper this result is extended to arbitrarily structured c grids an explicit formula is given for constructing an appropriate discretization of the coriolis terms the general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal c grid and the spherical longitude latitude c grid numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal pentagonal and triangular geodesic c grids on the sphere

solutions of fractional laplacian equations and their morse indices abstract in this paper we study the solutions of the following fractional laplacian equation s u f x u in u 0 in c under some assumptions on the nonlinearity we will show that the boundedness of solutions is equivalent to the finiteness of their morse indices we extend the result of bahri and lions 1992 1 from the laplacian equations to the fractional laplacian equations

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

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

multiple bound states of nonlinear schrodinger systems this paper is concerned with existence of bound states for schrodinger systems which have appeared as several models from mathematical physics we establish multiplicity results of bound states for both small and large interactions this is done by different approaches depending upon the sizes of the interaction parameters in the systems for small interactions we give a new approach to deal with multiple bound states the novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows for large interactions we use a minimax procedure to distinguish solutions by analyzing their morse indices

morse s oscillation theorem for a degenerate case abstract we consider the eigenvalue problem for a selfadjoint system of linear ordinary differential equations with general mixed boundary conditions which allow a combination of terms involving boundary values from the left and right end points we obtain a precise formula for the morse index of the problem which is valid even in the degenerate case

remarks on the resonant elliptic problem we consider the semilinear elliptic equation u f x u with the dirichlet boundary value for resonant nonlinearity at infinity using the morse theory and perturbation method some existence results of solution and multiple solutions are proved

periodic solutions of second order superquadratic hamiltonian systems with potential changing sign ii abstract we consider the periodic solutions of the second order hamiltonian system x x h t v x with v being positive and superquadratic at infinity h being c 1 2 periodic sign changing and all zeros being simple some existence and multiplicity results of periodic solutions are given

bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction abstract we study the global bifurcation curves of a diffusive logistic equation when harvesting is orthogonal to the first eigenfunction of the laplacian for values of the linear growth up to 2 examining in detail their behavior as the linear growth rate crosses the first two eigenvalues we observe some new behavior with regard to earlier works concerning this equation namely the bifurcation curves suffer a transformation at 1 they are compact above 1 there are precisely two families of degenerate solutions with morse index equal to zero and the whole set of solutions below 2 is not a two dimensional manifold

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition abstract in this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary under very general assumptions our main result says that each weak solution of such systems belongs to l l the proof is based on moser s iteration scheme the results presented here can also be applied to elliptic systems with homogeneous dirichlet boundary condition

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

morse theory for symmetric functionals on the sphere and an application to a bifurcation problem abstract this paper uses conley s index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group the authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite even if the action is not free this investigation has been motivated by the following bifurcation problem au lambda where a is a variational g equivariant operator an estimated on the number of branches bifurcating from an eigenvalue of a 0 is given author

characterization of solutions having finite morse index for some nonlinear pde with supercritical growth abstract in this paper we study solutions of the equation u u p in r n where n 2 n 2 p p c n see 3 for the definition of p c n we first classify all solutions not necessarily bounded having finite morse index for r n or r n with dirichlet boundary condition when is a regular bounded domain of r n for a family of solutions with dirichlet boundary condition u n we prove that u n l is bounded if and only if the morse index sequence i u n of u n is bounded finally we prove the same result for other nonlinearities f x u which have similar growth respectively at

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

liouville theorem for robin boundary value problems and finite morse indices in this paper we study the nonexistence result of the following nonlinear elliptic equation with robin boundary condition u u p 1uin r n u cu 0on r n under some assumptions on c and p we will show that this problem does not possess nontrivial bounded solution with finite morse index

superlinear indefinite elliptic problems and pohoz aev type identities abstract we prove the existence of a nontrivial solution for a nonlinear elliptic problem u u a x g u with dirichlet boundary condition on a bounded domain where g is superlinear both at zero and at infinity a x changes sign and 0

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

solutions of the mixed boundary problem and their morse indices abstract in this paper we study the solutions of the following mixed boundary problem u f x u in u 0 on 1 u 0 on 2 under some assumptions on the nonlinearity we will show that the boundedness of solutions is equivalent to the finiteness of their morse indices we extend the result of a bahri p l lions solutions of superlinear elliptic equations and their morse indices commun pure appl anal 45 1992 1205 1215 from the dirichlet boundary condition to the mixed boundary condition

nontrivial solutions of schrodinger equations with indefinite nonlinearities we prove the existence of nontrivial solutions for the schrodinger equation delta u v x u a gamma x f u in r n where f is superlinear and suberitical at zero and infinity respectively v is periodic and a x chances sign c 2007 elsevier inc all rights reserved

lyusternik schnirelman theory complementary principles and the morse index on considere une fonction a valeur reelle paire lisse sur une sphere de dimension finie s n genre et genre modulo 2 nombres critiques indices des points critiques non degeneres points critiques degeneres application a un probleme aux valeurs propres superlineaire

optimal mixed sensitivity for siso distributed plants the authors extend previous results for the single input single output siso mixed sensitivity problem for distributed parameter systems to a more general setting in which the transfer function of the plant to be controlled has both irrational stable and irrational unstable parts these developments were originally motivated by the computation of the infimal h sup spl infin norm of the weighted mixed sensitivity for the irrational transfer function model of a damped flexible beam