perturbation of doubly periodic solution branches with applications to the cahn hilliard equation abstract in this paper we prove the existence of doubly periodic solutions of certain nonlinear elliptic problems on r 2 and study the geometry of their nodal domains in particular we will show that if we perturb a nonlinear elliptic equation exhibiting a small amplitude doubly periodic solution whose nodal domains form a checkerboard pattern then the perturbed equation will have a unique nearby solution which is still doubly periodic but for which the nodal line structure breaks up moreover we indicate what can happen if we start with a large amplitude doubly periodic solution whose nodal domains form a checkerboard pattern and we relate these solutions to the cahn hilliard equation and spinodal decomposition

classification of convolutional codes convolutional codes have the structure of an f z module to study their properties it is desirable to classify them as the points of a certain algebraic variety by considering the correspondence of submodules and the points of certain quotient schemes and the inclusion of these as subvarieties of certain grassmannians one has a one to one correspondence of convolutional codes and the points of these subvarieties this classification of convolutional codes sheds light on their structure and proves to be helpful to give bounds on their free distance and to define convolutional codes with good parameters

on the number of positive solutions of weakly non linear elliptic equations when a parameter is large on etudie les solutions positives du probleme u f u dans u 0 sur ou est un domaine borne lisse de r n f r r est c 1 et est grand

every convolutional code is a goppa code convolutional goppa codes cgc were defined in appl algebra eng comm comput vol 15 pp 51 61 2004 and ieee trans inf theory vol 52 340 344 2006 in this paper we prove that every convolutional code is a cgc defined over a smooth curve over bbf q z and we give an explicit construction of convolutional codes as cgc over the projective line bbp bbf q z 1 we characterize which convolutional codes are defined by a complete linear system over curves of genus 0 1 and over hyperelliptic curves we apply these results to provide detailed constructions of some linear block codes as goppa codes

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

positivity properties of viscosity solutions of a degenerate parabolic equation we consider the problem i u t u u u 2 in r n r u x 0 u 0 x in r n where is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on r n

solutions of systems of elliptic differential equations on circular domains abstract we study global bifurcation of weak solutions of systems of elliptic differential equations considered on so 2 invariant domains we formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones as the main tool we use the degree for so 2 equivariant gradient maps defined by the second author in rybicki nonlinear anal tma 23 1 1994 83

two phase equilibria in the anti plane shear of an elastic solid with interfacial effects via global bifurcation we consider a simplified model of a two phase elastic solid with small interfacial energy in forced anti plane shear we propose a novel approach to finding meta stable equilibria local minima of the potential energy based upon global bifurcation theory a priori bounds and numerical path following in particular we treat the reciprocal of the capillarity coefficient as a continuation parameter and we fully exploit the hidden symmetry of our elliptic boundary value problem both strategies of which are crucial for obtaining rigorous existence results and for performing reliable and efficient computation we obtain branches of locally stable equilibria exhibiting phase nucleation and anti nucleation and fine layering of mixtures

bifurcations of the von karman equations with robin boundary conditions abstract we study buckling of a rectangular plate with elastic support and restraint along two edges and simple support along the other two edges this corresponds to a bifurcation analysis of the von karman equations with robin boundary conditions of special interest is the phenomenon of mode jumping in deformation of the plate we discretize the von karman equations with the finite difference methods the solution curves branching from the first two simple bifurcation points resulted from splitting of a double bifurcation point are numerically traced with respect to the increase of the load by using the continuation methods our numerical results show that mode jumping depends on length of the plate and stiffness of the elastic support

minimal positive solutions to some singular second order differential equations abstract the existence of a minimal c1 0 1 positive solution is established for some second order singular boundary value and initial value problems by new schemes which are related to x our nonlinearity may be singular at t 0 1 x 0 or x 0

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

unbounded sets of solutions of non cooperative elliptic systems on spheres abstract the aim of this paper is to show that any continuum of nontrivial solutions of a non cooperative system of elliptic equations on the sphere s n 1 bifurcating from the set of trivial solutions is unbounded moreover we characterize bifurcation points of this system at which the global symmetry breaking phenomenon occurs as the main tool we use the degree for so 2 invariant strongly indefinite functionals defined in 13

self similar intermediate asymptotics for a degenerate parabolic filtration absorption equation the equation tu u xx2u c 1 xu 2 is known in literature as a qualitative mathematical model of some biological phenomena here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock therefore we refer to this equation as a filtration absorption equation a family of self similar solutions to this equation is constructed numerical investigation of the evolution of non self similar solutions to the cauchy problems having compactly supported initial conditions is performed numerical experiments indicate that the self similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state identical zero an open problem caused by the nonuniqueness of the solution of the cauchy problem is discussed

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

singular boundary value problems for superlinear second order ordinary and delay differential equations abstract existence criteria are established for singular boundary value problems for nonlinear second order ordinary and delay differential equations the theorems obtained are very general and complement previous known results

preservation of the maslov index along bifurcating branches of solutions of first order systems in rn abstract in this paper we study bifurcation from simple eigenvalues for systems of differential equations we prove the existence of global bifurcating branches of solutions on which the maslov index of suitable associated linear systems is preserved

a class of one dimensional mds convolutional codes a class of one dimensional convolutional codes will be presented they are all mds codes i e have the largest distance among all one dimensional codes of the same length n and overall constraint length furthermore their extended row distances are computed and they increase with slope at least n in certain cases of the algebraic parameters we will also derive parity check matrices of vandermonde type for these codes finally cyclicity in the convolutional sense of 8 will be discussed for our class of codes it will turn out that they are cyclic if and only if the field element used in the generator matrix has order n this can be regarded as a generalization of the block code case

maximal entropy solutions under prescribed mass and energy abstract we consider a non local elliptic equation with exponential nonlinearity closely related to the mean field liouville equation the motivation for this equation is a variational entropy maximization problem under prescribed mass and energy we provide an unconditional existence proof in case of electrostatic repulsive self interaction and conditional existence and uniqueness in dimension two in the case of gravitational attractive self interaction

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

separation of global solution branches of elliptic systems with symmetry via nodal properties the idea that nodal properties are invariant along global solution branches of bifurcation problems goes back to the work of kolodner l and wolkowsky 2 31 in the context of nonlinear sturm liouville eigenvalue problems in that setting one can count the number of zeros of the solution a real valued function of a single real variable for the same class of problems it was shown later by crandall and rabinowitz 4 that this node count is inherited from the associated eigenfunction of the linearized problem about the trivial solution and that the invariance of nodal properties implies separation and unboundedness of global bifurcating solution branches it is natural to seek generalizations of these ideas to higher dimensions e g to problems involving more than one independent variable second order elliptic partial differential equations and or more than one dependent variable systems of second order differential equations these lead to immediate difficulties for elliptic partial differential equations we are confronted with the task of characterizing the set of zeros of functions of two or more variables in recent papers 5 6 we have shown how this problem simplifies considerably for equations with symmetries specifically in that setting we have demonstrated that the entire nodal set comprising onlyfixed hyper planes is precisely inherited from the eigenfunction of the linearization and is invariant along global branches of solutions possessing certain prescribed symmetries the proof requires groupinvariance arguments combined with subtle maximum principles in this work we take up the more challenging problem of systems of second order elliptic equations in the presence of symmetry to illustrate the difficulties involved we first discuss the case of a system of second order sturm liouville problems of the form

the computation of one parameter families of bifurcating elastic surfaces the problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms hat a alpha beta and hat b alpha beta is considered the undeformed shell is a spherical cap of radius r and thickness h with an angular width 2 theta 0 where 0 theta 0 pi 2 the cap is subjected to a constant uniform load lambda and is simply supported at its edge the authors seek to compute the one parameter families of buckled states that branch from the unbuckled state of the shell this is accomplished in two steps first a finite element method is used to solve the governing shell equations a pair of fourth order nonlinear partial differential equations pdes a solution of this system is a curvature potential w a stress potential f and the load lambda using liapunov schmidt reduction it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell in the work presented here these

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

nonradial maximizers for a henon type problem and symmetry breaking bifurcations for a liouville gel fand problem with a vanishing coefficient we study the radiality and nonradiality of the maximizers of begin aligned sup left int d x alpha e u 1 dx u in h 0 1 d int d left nabla u right 2dx beta 2 right end aligned from a viewpoint of the bifurcation diagram of the euler lagrange equation begin aligned delta u beta 2 frac x alpha e u int d x alpha ue udx 0 hbox in d qquad u 0 hbox on partial d end aligned where d x in mathbb r 2 x 1 we show that the maximizers are nonradial if begin aligned beta sqrt 8 pi left alpha 2 log frac 2 alpha 4 alpha frac alpha 4 2 right end aligned and that the maximizers are radial if alpha 0 or beta 0 is small we also give the bifurcation diagrams of the liouville gel fand problem with a vanishing coefficient begin aligned delta u lambda x alpha e u 0 hbox in d qquad u 0 hbox on partial d end aligned and of the associated mean field equation the analysis is based on exact solutions and explicit calculations

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

existence of positive solutions for the one dimensional singular p laplacian this paper concerns the positive solutions of boundary value problems for the one dimensional singular p laplacian by the classical method of elliptic regularization we obtain some existence results which generalize some results of w zhou x wei positive solutions to bvps for a singular differential equation nonlinear anal 2006 doi 10 1016 j na 2006 06 015

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

discontinuous viscosity solutions of a degenerate parabolic equation we study a nonlinear degenerate parabolic equation of the second order regularizing the equation by adding some artificial viscosity we construct a generalized solution we show that this solution is not necessarily continuous at all points

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

global bifurcations of solutions of elliptic differential equations abstract the aim of this article is to prove global bifurcation theorems for s 1 equivariant potential operators of the form compact perturbation of identity as an application we prove that components of the set of nontrivial solutions of system formula which bifurcate from the set of trivial solutions are unbounded in suitably chosen space the important point to note here is that our results apply only to systems with an even number of equations as the main tool we use degree theory for s 1 equivariant potential operators constructed by the author in 16 and developed in 17

monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions abstract let b r n n 1 be a unit ball and d r m be a bounded domain we study the global branch consisting of interior single peak solutions of the elliptic neumann problem u u u p 0 in b u 0 on b and the monotonicity of the first eigenvalue along the branch for large where 1 p p s n 2 n 2 n 3 n 1 2 when the domain is replaced with b d using this monotonicity we show that the branch of solutions concentrating on 0 d has secondary bifurcation points for n 1 and p 2 an integer we determine the global bifurcation diagram by showing the monotonicity of the time map the period function the monotonicity of the first eigenvalue along the whole branch is also proved for n 1 and p 3

toric geometry and equivariant bifurcations abstract many problems in equivariant bifurcation theory involve the computation of invariant functions and equivariant mappings for the action of a torus group we discuss general methods for finding these based on some elementary considerations related to toric geometry a powerful technique in algebraic geometry this approach leads to interesting combinatorial questions about cones in lattices which lead to explicit calculations of minimal generating sets of invariants from which the equivariants are easily deduced we also describe the computation of hilbert series for torus invariants and equivariants within the same combinatorial framework as an example we apply these methods to the interaction of two linear modes of a euclidean invariant pde on a rectangular domain with periodic boundary conditions

pattern formation of the stationary cahn hilliard model we investigate critical points of the free energy of the cahn hilliard model of a binary alloy under the constraint of a constant mass the domain is the unit square minimizers of the energy without interfacial energy term are given by a decomposition of the two components of the alloy but the interfaces between the components are arbitrary specic patterns are only formed if an interfacial energy term is present we select such patterns of minimizers by an approximation of se quences of conditionally critical points of the free energy when the interfacial energy term tends to zero this is what we call pattern formation of the stationary cahn hilliard model mathematically it is a singular limit process we obtain the conditionally critical points by a global bifurcation analysis of the euler lagrange equation for the free energy where the mass is the bifurcation pa rameter and where the constant homogeneous mixtures give the trivial solutions by using characteristic symmetries and monotonicities of the bifurcating solutions we show that singular limits exist for all masses in the so called spinodal region and that they are minimizers of the free energy without interfacial energy term a binary alloy in a vessel can be described by a function u ir as follows u x 2 0 1 means that the mixture contains u x 100 of one component at x2 the energy density of the alloy is modelled by w u where w is a two well potential figure 1

solutions of nonlinear planar elliptic problems with triangle symmetry abstract in this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in fifeet al physica d100 1997 257 278 more precisely we consider small amplitude solutions of u f u 0 in r 2whose nodal domains consist of equilateral triangles tiling the plane if this equation is suitably perturbed then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up furthermore we treat the non generic rectangular cases which had to be excluded in fifeet al physica d100 1997 257 278 as well as other nodal domain structures

local structure of bifurcation curves for nonlinear sturm liouville problems abstract we consider the nonlinear bifurcation problem arising in population dynamics and nonlinear schrodinger equation u t f u t u t 0 t i 0 1 u 0 u 1 0 where 0 is a parameter we mainly treat the case where f u u u p p 1 and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point 2 in l q framework together with the result of the global behavior of the bifurcation curve we understand completely the structure of the bifurcation curve we also consider the nodal solution u n of the equation u t u t u t p 1 u t with u 0 u 0 and establish an asymptotic expansion formula for with respect to the gradient norm of the solution associated with as n 2

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

positive solutions to bvps for a singular differential equation abstract in this paper we are interested in the positive solutions to a singular differential equation 2 f t 0 0 t 1 subject to the first or all of the following boundary conditions 1 0 0 1 0 0 where is a positive constant f t c 1 0 1 f t 0 on 0 1 by the classical method of elliptic regularization some existence results of the above bvps are obtained moreover the uniqueness and the asymptotic behaviors as 0 and as will also be studied

scaling solution branches of one parameter bifurcation problems we consider an algorithm for analyzing bifurcation structure and for branch switching in solution branches of one parameter dependent problems based on liapunov schmidt methods and analyses of scales of solutions we verify the existence of bifurcating solution branches successively via truncated taylor expansions of the reduced bifurcation equations and an enlarged system as examples bifurcations of semilinear elliptic differential equations are studied on square and hexagon domains

bifurcation and symmetry breaking of solutions of systems of elliptic differential equations in this article we study global bifurcations of weak solutions of the following variational system of elliptic differential equations u au u u in u 0on we prove sufficient conditions for the existence of unbounded continua of nontrivial solutions emanating from the trivial ones and necessary conditions for the existence of bounded continua moreover we describe a symmetry breaking phenomenon that occurs on that continua

asymptotic oscillations of continua of positive solutions of a semilinear sturm liouville problem abstract we consider the set of positive solutions u of the semilinear sturm liouville boundary value problem u u f u in 0 u 0 u 0 where f 0 r is lipschitz continuous and is a real parameter we suppose that f s oscillates as s in such a manner that the problem is not linearizable at u but does nevertheless have a continuum c of positive solutions bifurcating from infinity we investigate the relationship between the oscillations of f and those of c in the u 0 plane at large u 0 in particular we discuss whether c oscillates infinitely often over a single point or over an interval i of positive length of values an immediate consequence of such oscillations over i is the existence of infinitely many solutions of arbitrarily large norm u 0 of the problem for all values of i

branches of positive solutions of quasilinear elliptic equations on non smooth domains abstract we prove existence of an unbounded global branch i e connected set of weak solutions of a second order quasilinear equation depending on a real parameter on an arbitrary possibly non smooth bounded domain in r n with a leray lions operator as the leading part here we can allow lower order nonlinearities which depend on first derivatives satisfying appropriate growth conditions including the critical case furthermore we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive our approach also yields a new existence result in the case of critical growth in derivatives of lower order

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

lq inverse spectral problems for semilinear sturm liouville problems abstract we consider the bifurcation problem u t f u t u t u t 0 t i 0 1 u 0 u 1 0 where 0 is a positive parameter we determine the nonlinear term f u by the asymptotic behavior of the bifurcation curve of the positive solutions in r l q i where 1 q

convolutional goppa codes in this correspondence we define convolutional goppa codes over algebraic curves and construct their corresponding dual codes examples over the projective line and over elliptic curves are described obtaining in particular some maximum distance separable mds convolutional codes

global bifurcation of positive solutions from zero in nonlinearizable elliptic problems with indefinite weight abstract this paper is devoted to the study of global bifurcation of solutions from zero of some nonlinearizable eigenvalue problems for second order elliptic partial differential equations with indefinite weight the existence of global continua of solutions emanating from bifurcation intervals surrounding the trivial solutions corresponding to the principal eigenvalues of the linear problem obtained by setting the nonlinear term equal to zero and contained in the classes of positive and negative functions is shown

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains keywords semilinear second order elliptic equation dirichlet boundary condition unbounded branch of positive weak solutions reference ana article 1994 001view record in web of science record created on 2008 12 10 modified on 2016 08 08