the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

darboux transformations of jacobi matrices and pade approximation let j be a monic jacobi matrix associated with the cauchy transform f of a probability measure we construct a pair of the lower and upper triangular block matrices l and u such that j lu and the matrix jc ul is a monic generalized jacobi matrix associated with the function fc f 1 it turns out that the christoffel transformation jc of a bounded monic jacobi matrix j can be unbounded this phenomenon is shown to be related to the effect of accumulating at of the poles of the pade approximants of the function fc although fc is holomorphic at the case of the ul factorization of j is considered as well

meromorphic quadratic differentials and measured foliations on a riemann surface we describe the space of measured foliations induced on a compact riemann surface by meromorphic quadratic differentials we prove that any such foliation is realized by a unique such differential q if we prescribe in addition the principal parts of sqrt q at the poles this generalizes a theorem of hubbard and masur for holomorphic quadratic differentials the proof analyzes infinite energy harmonic maps from the riemann surface to mathbb r trees of infinite co diameter with prescribed behavior at the poles

a detailed analysis on local bifurcation from infinity for nonlinear elliptic problems in this paper we analyze the local side of the bifurcation from infinity at the first eigenvalue of several elliptic operators we underline that the key to decide the local behavior of the bifurcation lies in the sign of certain integral involving all the values of the nonlinearity

spectral problems for generalized jacobi matrices a new class of generalized jacobi matrices is introduced every proper real rational function is proved to be the m function of a unique finite generalized jacobi matrix moreover every generalized nevanlinna function m which is a solution of a determinate indefinite moment problem turns out to be the m function of a unique infinite generalized jacobi matrix the method we use is based on the step by step schur process of solving the indefinite moment problem the convergence of the sequence of subdiagonal pade approximants for the corresponding hamburger series is investigated

landesman lazer conditions at half eigenvalues of the p laplacian abstract we study the existence of solutions of the dirichlet problem 1 p u a p u a p u p u f x u x 0 1 2 u 0 u 1 0 where p 1 p s s p 1 sgn s for s r the coefficients a c 0 0 1 r and u max u 0 we suppose that f c 1 0 1 r and that there exist f c 0 0 1 such that lim f x f x for all x 0 1 with these conditions the problem 1 2 is said to have a jumping nonlinearity we also suppose that the problem 3 p u a p u a p u p u on 0 1 together with 2 has a non trivial solution u that is is a half eigenvalue of 2 3 and the problem 1 2 is said to be resonant combining a shooting method with so called landesman lazer conditions we show that the problem 1 2 has a solution most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients a are constants and the resonance has been at a point in the fucik spectrum even in this constant coefficient case our result extends previous results in particular previous variational approaches have required strong conditions on the location of the resonant point whereas our result applies to any point in the fucik spectrum

the fredholm alternative at the first eigenvalue for the one dimensionalp laplacian abstract in this work we study the range of the operatoru u p 2 u 1 u p 2 u u 0 u t 0 p 1 we prove that all functionsh c1 0 t satisfying t0 h t sinp pt t dt 0 lie in the range but that ifp 2 andh 0 the solution set is bounded here sin pt t is a first eigenfunction associated to 1 we also show that in this case the associated energy functionalu 1 p t0 u p 1 p t0 u p t0 huis unbounded from below if 1 2 onw1 p0 0 t 1corresponds precisely to the best constant in thelp poincare inequality moreover we show that unlike the linear casep 2 forp 2 the range contains a nonempty open set inl 0 t

disappearance of cusp forms in special families in his 1954 g6ttingen lectures atle selberg sb posed the question of the asymptotic density of the poles of the eisenstein series for r a cofinite non cocompact subgroup of psl 2 r the question is known to be equivalent to the asymptotic count of the number of eigenvalues for the laplacebeltrami operator d acting in l2 h r where h is the upper halfplane with hyperbolic metric in general the spectral decomposition of d consists of the continuous band 1 4 oo with multiplicity the number of cusps of h r and a discrete component possibly in 1 4 0o spanned by eigenfunctions w1i wp2 see hj1 kb lxp for the eigenvalues aj dpj ajpj o consider the spectral counting function n r ij v n 0 see dips ps2 cor 2 3 luo lu recently established the unconditional result his argument does not involve the lindel6f hypothesis deshouillers et

generalized jacobi operators in krein spaces a special class of generalized jacobi operators which are self adjoint in krein spaces is presented a description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given in particular special attention is paid to the periodic generalized jacobi operators finally the spectral properties of generalized jacobi operators are applied to prove convergence results for pade approximants

unbounded nonlinear perturbations of linear elliptic problems at resonance 435 where lj is an eigenvalue of l a is the derivative in the direction outward normal to the boundary 2 of 0 and f h q recently shapiro lo established an existence result for the problem 1 1 in the case that g has unbounded nonlinearities and satisfies a landesman lazer type condition shapiro s result however does not cover the case that g has linear growths our first result is an extension of shapiro s result to the case that g has a linear growth we next consider the case g does not satisfies any landesman lazer type condition that is we study the case lim inf oc g x t lim supi up g x t a e on q several authors considered this case with g having bounded nonlinearities cf 9 fucik and krebec 5 and hess 6 gave existence results for 1 1 in case that j l q and g is a continuous function satisfying g x t g t g t g r and g lim fm g t 0 we extend these results for 1 1 to the case g has unbounded nonlinearities and f l2 our method employed here is different from those in s 6 or lo 2

moduli spaces of higgs bundles on degenerating riemann surfaces abstract we prove a gluing theorem for solutions a 0 0 of hitchin s self duality equations with logarithmic singularities on a rank 2 vector bundle over a noded riemann surface 0 representing a boundary point of teichmuller moduli space we show that every nearby smooth riemann surface 1 carries a smooth solution a 1 1 of the self duality equations which may be viewed as a desingularization of a 0 0

landesman lazer conditions for resonant p laplacian problems with jumping nonlinearities we study the existence of solutions of the p laplacian dirichlet problem 1 p u p u h x u f x u u x 0 1 2 u 0 u 1 0 where r p 1 s s 1sgn for s 1 r the function h 0 1 r r has the form h x a x p a x p x 0 1 r with max 0 and a l1 0 1 and the function f 0 1 r2 r is continuous and satisfies f x k x 1 q 1 x 0 1 r2 for some q 1 p and k l1 0 1 the dominant asymptotic behaviour of equation 1 as u is determined by the coefficients a and we allow a a in which case the problem is said to be jumping if the positively homogeneous problem obtained from 1 2 by setting f 0 has a non trivial solution then the problem is said to be resonant and is said to be a half eigenvalue assuming that the problem 1 2 is both jumping and resonant we will obtain a solution under certain landesman lazer conditions on f

two point boundary value problems for second order ordinary differential equations across many resonant points abstract several optimality results on the existence and uniqueness of solutions to two point boundary value problems across many resonant points y x y 0 y 0 y0 y 1 y1 are proved by the optimal control theory method

on global dynamics of reaction diffusion systems at resonance abstract in this paper we use the homotopy invariants methods to study the global dynamics of the reaction diffusion systems that are at resonance at infinity considering degrees of the resonance for the nonlinear perturbation we establish landesman lazer type conditions and use them to express the rybakowski conley index of the invariant set consisting of all bounded solutions obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations

solvability of the resonant 1 dimensional periodic p laplacian equations abstract in this paper we study the existence of 2 periodic solutions of x p 2 x x p 2 x f t x r where p 1 is an eigenvalue of the corresponding homogeneous equation f is a 2 periodic function some existence results are proved via the morse theory

the asymptotic behavior of green s functions for degenerating hyperbolic surfaces given a degenerating family of riemann surfaces of genus g 2 it is a natural and important question to study their spectral behaviors in particular green s functions during degeneration see 9 10 17 11 13 and 5 by the uniformization theorem any riemann surface of genus g 2 has a canonical metric of constant curvature 1 from now on we call a riemann surface with this canonical metric a hyperbolic surface and a degenerating family of hyperbolic surfaces is denoted by st l 0 the degeneration of st is caused by the pinching of several simple closed geodesics 71 l 7 1 on st that is the lengths of these geodesics go to zero as l 0 let at be the laplacian of st then the kernel of at consists of functions which are constant on each connected component of st and at is invertible on the orthogonal complement of the kernel the schwartz kernel of this inverse is called the green function of st and denoted by g z w t for z w sl we would like to study the asymptotic behavior of g z w i as l o in order to compare functions on s o and st we use the harmonic map of infinite energy so s 71 l 7 l constructed by wolf el4 intuitively speaking this map z opens up each node a pair of cusps of so into a simple closed pinching geodesic of sf at first we assume that there is only one pinching geodesic l 7 of length l on s if 7 separates st then so has two components se and so z of genus g gz respectively

oscillation and interlacing for various spectra of the p laplacian abstract we consider the eigenvalue problem p u r q u p 1 sgn u a e on 0 b where b 0 p 1 p is the p laplacian q r l 1 0 b r 0 and r a variety of boundary conditions will be imposed at 0 and b but the main focus is on dirichlet neumann and periodic anti periodic conditions in the linear case p 2 there are well known oscillation and interlacing results for the eigenvalues of the above problem with these boundary conditions and we explore similarities and differences between this case and the nonlinear case with p 2 we will see that several new phenomena occur in the periodic anti periodic problems for example with separated boundary conditions the structure of the spectrum when p 2 is the same as when p 2 but this is not true in the periodic anti periodic cases we also consider the set of half eigenvalues and the fucik spectrum of the problem these concepts are useful for solving jumping nonlinearity problems and we show that new phenomena also appear here in addition to analogues of those occurring for the usual eigenvalues

resonance with respect to the fucik spectrum for non selfadjoint operators abstract in this paper we consider a real linear non selfadjoint operator l dom l l 2 l 2 and study the solvability of the problem l u u u g u f 0 in the resonance case with respect to the fucik spectrum l we extend the standard landesman lazer type conditions and compare our result with results for selfadjoint operators as well as with results for the resonance in the point spectrum i e for l

inverse problems for periodic generalized jacobi matrices abstract some inverse problems for semi infinite periodic generalized jacobi matrices are considered in particular a generalization of the abel criterion is presented the approach is based on the fact that the solvability of the pell abel equation is equivalent to the existence of a certainly normalized j unitary 2 2 matrix polynomial the monodromy matrix

on the resonance problem with nonlinearity which has arbitrary linear growth abstract this paper deals with the nonlinear two point boundary value problem at resonance even nonlinearities g with an arbitrary linear growth in resp may be considered but only on the cost of the corresponding bound on their linear growth at resp it generalizes the previous results in this direction obtained by m schechter j shapiro and m snow trans amer math soc 241 1978 69 78 l cesari and r kannan proc amer math soc 88 1983 605 613 and s ahmad proc amer math soc 93 1984 381 384

perturbation of the p laplacian by vanishing nonlinearities in one dimension abstract we treat the quasi linear spectral problem p u p 2 u u p 2 u h x u 0 x a u 0 u a 0 at resonance with vanishing nonlinearity h x u g u f x g u 0 as u and f l 0 a f 0 satisfying certain orthogonality related hypotheses 1 p 3 p 2 the parameter r takes an arbitrary resonant value we first establish the boundedness of the set of all weak solutions in the sobolev space w 0 1 p 0 a which then enables us to obtain an existence result by the leray schauder degree theory the boundedness is obtained from a very precise asymptotic estimate valid for large solutions to p which can be applied thanks to a sufficiently fast rate of decay g u 0 as u

solvability of a boundary value problem with the nonlinearity satisfying a sign condition soit f 0 r r une fonction donnee satisfaisant des conditions de caratheodory et h 0 r une fonction donnee dans l 1 0 on obtient l existence d une solution au probleme aux valeurs limites u x u x f x u x h x x 0 u 0 u 0 pour h x telle que 0 h x sinxdx 0 et f x u telle que f x u u 0 pour u r et il y a une constante 0 telle que limsup u f x u u 3 uniformement pour x 0

second order sturm liouville problems with asymmetric superlinear nonlinearities abstract we consider the nonlinear sturm liouville problem p x u x q x u x f x u x h x in 0 c 00 u 0 c 01 u 0 0 c 10 u c 11 u 0 where p c 1 0 q c 0 0 with p x 0 for all x 0 ci02 ci12 0 i 0 1 h l2 0 we suppose that f 0 r r is continuous and there exist increasing functions l u 0 r and positive constants a b such that lim t l t and a l f x a u 0 f x a b 0 for all x 0 thus the nonlinearity is superlinear as u x but linearly bounded as u x existence and non existence results are obtained for the above problem similar results have been obtained before for problems in which f is linearly bounded as and these results have been expressed in terms of half eigenvalues of the problem the results obtained here for the superlinear case are expressed in terms of certain asymptotes of these half eigenvalues

harmonic maps from degenerating riemann surfaces we study harmonic maps from degenerating riemann surfaces with uniformly bounded energy and show the so called generalized energy identity we find conditions that are both necessary and sufficient for the compactness in w 1 2 and c 0 modulo bubbles of sequences of such maps

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

index theory nontrivial solutions and asymptotically linear second order hamiltonian systems abstract in this paper we consider the existence and multiplicity of solutions of second order hamiltonian systems we propose a generalized asymptotically linear condition on the gradient of hamiltonian function classify the linear hamiltonian systems prove the monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear hamiltonian systems by global analysis methods such as the leray schauder degree theory the morse theory the ljusternik schnirelman theory etc

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

nonlinear boundary value problems at resonance soit un espace de mesure fini et h l 2 l espace des fonctions a valeurs reelles de carre integrable mesurables de lebesgue sur on considere l equation lu r n u g u h ou h h l dom l h h est un operateur lineaire auto adjoint densement defini r n appartient au spectre l g r r est une fonction qui peut etre non lineaire on donne des resultats d existence generaux

the grafting map of teichmuller space one of the underlying principles in the study of kleinian groups is that aspects of the complex projective geometry of quotients of by these groups reflect properties of the three dimensional hyperbolic geometry of the quotients ofh by these groups yet even though it has been over thirty five years since lipman bers exhibited a holomorphic embedding of the teichmuller space of riemann surfaces in terms of the projective geometry of a teichmuller space of quasi fuchsian manifolds no corresponding embedding in terms of the three dimensional hyperbolic geometry has been presented one of the goals of this paper is to give such an embedding this embedding is straightforward and has been expected for some time ta97 mc98 to each member of a bers slice of the space qf of quasi fuchsian 3manifolds we associate the bending measured lamination of the convex hull facing the fixed conformal end the geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting an operation on projective structures on surfaces that traces its roots back at least to klein kl33 p 230 50 with a modern history developed by many authors ma69 he75 fa83 st83 go87 gkm00 ta97 mc98 the main technical tool in our proof that bending measures give coordinates for bers slices and the second major goal of this paper is the completion of the proof of the grafting conjecture this conjecture states that for a fixed measured lamination the self map of teichmuller space induced by grafting a surface along is a homeomorphism of teichmuller space our contribution to this argument is a proof of the injectivity of the grafting map while the principal application of this result that we give is to geometric coordinates on the bers slice of qf one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of kleinian groups see 5 2 thus we feel that this result is of interest in its own right we now state our results and methods more precisely throughout s will denote a fixed differentiable surface which is closed orientable and of genus g 2 let tg be the corresponding teichmuller space of marked conformal structures on s and let pg denote the deformation space of complex projective structures on s see 2 for definitions

on the fredholm alternative for the p laplacian at higher eigenvalues in one dimension abstract we investigate the existence of a weak solution u to the quasilinear two point boundary value problem p u p 2 u k u p 2 u f x 0 x a u 0 u a 0 we assume 1 p 2 0 a and f l 0 a is a given function f 0 the number k stands for the k th eigenvalue of the one dimensional p laplacian let sin p p x a denote the eigenfunction associated with 1 then sin p k p x a is the eigenfunction associated with k it is well known that problem p has a solution when k 1 and f satisfies the orthogonality condition 0 a f x sin p k p x a d x 0 we focus on the case k 2 assuming the orthogonality condition above we obtain new existence results for p which significantly generalize those already known manasevich and p takac 2002 4 we are able to handle more degenerate sufficient conditions on f than any previous work we present a simple example of f to which previous results and methods are not applicable our approach is based on the third and second order taylor expansions of the functions sin p and cos p sin p respectively in the l 1 norm rather than in the standard l norm

bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity this paper deals with existence results for the following nonlinear problem with the dirichlet p laplacian p in a bounded domain rn pu u p 2u h x u in u 0on here pu defdiv u p 2 u where p 1 is a fixed number h h x s is a given function from r into r and r stands for a spectral parameter we focus on close to 1 including the resonant case 1 the nonlinearity h is assumed to be of landesman lazer type but we can deal with vanishing nonlinearities as well our asymptotic method substitutes the lyapunov schmidt method in some sense unlike in the semilinear case p 2 our method can treat more general nonlinearities if p 2 vanishing nonlinearities with very fast decay

harmonic maps into hyperbolic 3 manifolds high energy degeneration of harmonic maps of riemann surfaces into a hyperbolic 3 manifold target is studied generalizing results of m 1 in which the target is two dimensional the hopf foliation of a high energy map is mapped to an approximation of its geodesic representative in the target and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy the images of harmonic maps obtained when the domain degenerates along a teichmuller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

unbounded perturbations of forced second order ordinary differential equations at resonance we prove the existence of at least one solution for the differential equation x t m2x t g t x t e t with periodicity conditions x 0 x 2 x 0 x 2 0 where m 0 is an integer e is integrable and g satisfies caratheodory conditions our results are obtained for the case when there is resonance at the eigenvalue m2 of the linear second order differential equation t x t 0 r with x 0 x 2 x 0 x 2 0 the function g may be unbounded and touching of the eigenvalue m 1 2 resp m 1 2 if m 0 on a subset of positive measure is allowed our approach also works when periodicity conditions are replaced by dirichlet or neumann boundary conditions the proofs are based on topological degree mawhin s continuation theorem and leray schauder techniques

the asymptotic behavior of the takhtajan zograf metric we obtain the asymptotic behavior of the takhtajan zograf metric on the teichmuller space of punctured riemann surfaces

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

harmonic maps between surfaces and teichmuller spaces we study harmonic maps between two distinct compact riemann surfaces of the same genus our aim is to examine the role of the curvature in the relationship between energies of harmonic maps and conformal metrics on the target surface we consider the energy of the energy minimizing map between the surfaces as a function of the metrics within the conformal class of the target and solve the variational problem of minimizing this functional when we restrict the upper curvature bound and normalize the area of the metrics in doing so we show that the upper curvature bound on the normalized target metric is an obstruction in obtaining a degree 1 conformal harmonic map between distinct riemann surfaces

non resonance conditions for semilinear sturm liouville problems with jumping non linearities abstract we consider the sturm liouville boundary value problem p x u x q x u x f x u x h x x 0 c00u 0 c01u 0 0 c10u c11u 0 where p c1 0 q c0 0 with p x 0 x 0 c2i0 c2i1 0 i 0 1 h l2 0 and f 0 r r is a caratheodory function we assume that the rate of growth of f x is at most linear as but the asymptotic behaviour may be different as so the non linearity is termed jumping conditions for existence of solutions of this problem are usually expressed in terms of non resonance with respect to the standard fucik spectrum in this paper we give conditions for both existence and non existence of solutions in terms of a slightly different idea of the spectrum these conditions extend the usual fucik spectrum conditions

the fredholm alternative for the one dimensional p laplacian abstract in this paper we consider the solvability of the boundary value problem p u 1 p u f u h t u 0 u t 0 where p 1 p u u p 2 u by using generalized polar coordinates transformation method we improve and generalize some results obtained recently in j differential equations 151 1999 386

the rotation number approach to the periodic fuc caron ik spectrum abstract in this paper we study the fucik spectrum of the problem q t x q t x 0 with the 2 periodic boundary condition where q t are 2 periodic after introducing a rotation number function for we prove using the hamiltonian structure and the positive homogeneity of that for any positive integer n the two boundary curves of the domain 1 n 2 in the plane are fucik curves of the result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional fucik spectrum with the dirichlet condition in particular it remains open if the fucik spectrum of is composed of only these curves

the fucik spectrum of general sturm liouville problems abstract consider the boundary value problem pu qu u u in 0 c 00 u 0 c 01 u 0 0 c 10 u c 11 u 0 where u max u 0 the set of points r 2 for which this problem has a non trivial solution is called the fucik spectrum when p 1 q 0 and either dirichlet or periodic boundary conditions are imposed the fucik spectrum is known explicitly and consists of a countable collection of curves with certain geometric properties in this paper we show that similar properties hold for the general problem above and also for a further generalization of the fucik spectrum we also discuss some spectral type properties of a positively homogeneous half linear problem and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities

bifurcation surfaces stemming from the fucik spectrum abstract let r n be a bounded domain with smooth boundary and g r r be a nonlinear function we prove existence of two dimensional bifurcation surfaces for the elliptic boundary value problem u a u b u g x u in u 0 where u min 0 u u max 0 u and a b r 2 is a pair of parameters we show that these two dimensional bifurcation surfaces stem from the fucik spectrum of the main difficulty in doing that comes from non smoothness of the operators u u in order to overcome this difficulty a variant implicit function theorem and an abstract two dimensional bifurcation theorem are proved these two theorems do not require smoothness of operators and the abstract two dimensional bifurcation theorem can be regarded as an extension of the well known crandall rabinowitz bifurcation theorem and therefore are of interest for their own sake

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

half eigenvalues and the fucik spectrum of multi point boundary value problems abstract we consider the nonlinear boundary value problem consisting of the equation 1 u f u h a e on 1 1 where h l 1 1 1 together with the multi point dirichlet type boundary conditions 2 u 1 i 1 m i u i where m 1 are integers 1 m 0 1 m 1 1 m and we suppose that i 1 m i 1 we also suppose that f r r is continuous and 0 f lim s f s s we assume that these limits exist we allow f f such a nonlinearity f is said to be jumping related to 1 is the equation 3 u a u b u on 1 1 where a b 0 and u x max u x 0 for x 1 1 the problem 2 3 is positively homogeneous and jumping regarding a b as fixed values of a b for which 2 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem which we briefly describe equivalent solvability and non solvability results for 1 2 are obtained from either the half eigenvalue or the fucik spectrum approach

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

second order three point boundary value problems with jumping non linearities abstract we consider the non linear three point boundary value problem consisting of the equation 1 u f u h a e on 0 1 where h l 1 0 1 together with the boundary conditions 2 u 0 0 u 1 u where 0 1 the function f r r is continuous and we assume that the following limits exist and are finite f lim s f s s we allow f f such a non linearity f is said to be jumping related to 1 is the equation 3 u a u b u u on 0 1 where a b r and u x max u x 0 for x 0 1 the problem 2 and 3 is positively homogeneous and jumping values of for which 2 and 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 and 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability and non solvability results for 1 and 2 are obtained from either the half eigenvalue or the fucik spectrum approach

the picard boundary value problem for nonlinear second order vector differential equations the existence of solutions is studied for the picard boundary value problem b x b f b t b x b x 0 b x b a b x b b 0 where b b b a and b f b a b b b r sup n b r sup n rarr b r sup n is a continuous function

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

grafting hyperbolic metrics and eisenstein series the family hyperbolic metric for the plumbing variety zw t and the non holomorphic eisenstein series e zeta 2 are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of riemann surfaces applications include an asymptotic expansion for the weil petersson metric and a local form of symplectic reduction