hyponormality of toeplitz operators on the weighted bergman space with matrix valued circulant symbols abstract in this paper we consider the hyponormality of toeplitz operators t on the weighted bergman space with matrix valued circulant symbols we give a necessary and sufficient condition for the hyponormality of t with such a symbol under some constraint on the entries of the matrix

subnormal and quasinormal toeplitz operators with matrix valued rational symbols abstract in this paper we deal with the subnormality and the quasinormality of toeplitz operators with matrix valued rational symbols in particular in view of halmos s problem 5 we focus on the question which subnormal toeplitz operators are normal or analytic we first prove let l m n be a matrix valued rational function having a matrix pole i e there exists d for which ker h z h c n 2 where h denotes the hankel operator with symbol if i t is hyponormal ii ker t t is invariant for t then t is normal hence in particular if t is subnormal then t is normal next we show that every pure quasinormal toeplitz operator with a matrix valued rational symbol is unitarily equivalent to an analytic toeplitz operator

vicious walkers and hook young tableaux we consider a generalization of the vicious walker model using a bijection map between the path configuration of the non intersecting random walkers and the hook young diagram we compute the probability concerning the number of movements of the walker applying the saddle point method we reveal that the scaling limit gives the tracy widom distribution which is the same with the limit distribution of the largest eigenvalues of the gaussian unitary ensemble

harnack inequalities for solutions of general second order parabolic equations and estimates of their holder constants on etudie les solutions u des equations lineaires lu d 0 u i j n a ij d i d j u i 1 n bid i u cu f et des equations non lineaires mu d 0 u i j n a ij d 0 d j u a u du 0 dans l espace de sobolev w 1 2 n 1 1 2 q on deduit une inegalite de harnack modifiee pour les solutions

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

geometric properties of boundary sections of solutions to the monge ampere equation and applications abstract in this paper we establish several geometric properties of boundary sections of convex solutions to the monge ampere equation the engulfing and separating properties and volume estimates as applications we prove a covering lemma of besicovitch type a covering theorem and a strong type p p estimate for the maximal function corresponding to boundary sections moreover we show that the monge ampere setting forms a space of homogeneous type

linear oblique derivative problems for the uniformly elliptic hamilton jacobi bellman equation on demontre la resolubilite et l unicite des problemes aux valeurs limites generaux et de dirichlet

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

on a class of jointly hyponormal toeplitz operators we characterize when a pair of toeplitz operators t t t is jointly hyponormal under various assumptions for example is analytic or is a trigonometric polynomial or is analytic a typical characterization states that t t t is jointly hyponormal if and only if an algebraic relation of and holds and the single toeplitz operator t is hyponormal where is a combination of and more general results for an n tuple of toeplitz operators are also obtained

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

oblique boundary value problems for augmented hessian equations ii abstract in this paper we continue our investigations into the global theory of oblique boundary value problems for augmented hessian equations we construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented hessian is only assumed regular this enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case finally we give some applications to existence theorems which embrace standard hessian equations as special cases

toeplitz algebras and spectral results for the one dimensional heisenberg model we determine the structure of the spectrum and obtain nonpropagation estimates for a class of toeplitz operators acting on a subset of the lattice zn this class contains the hamiltonian of the one dimensional heisenberg model

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

intermediate schauder theory for second order parabolic equations i estimates on introduit les normes et les espaces de fonctions a utiliser les estimations a priori pour les problemes a coefficients constants dans des domaines simples sont reduits a des estimations integrales on donne des conditions sous lesquelles on peut passer localement a des coefficients variables on donne des estimations globales

hyponormal toeplitz operators and extremal problems of hardy spaces the symbols of hyponormal toeplitz operators are completely described and those are also studied being related with the extremal problems of hardy spaces moreover we discuss halmos s question about a subnormal toeplitz operator when the self commutator is finite rank

on existence of solutions for a class of fully nonlinear noncoercive problems 0 arises in the three dimensional water wave theory see m shinbrot lo and also in the engineering problem of hydraulic fracturing of oil wells see j r canon and g h meyer 2 7 mathematically the model problem 1 is of interest since it represents one of the simplest noncoercive elliptic problems i e the lopatinski shapiro condition fails see 4 for which there is no general theory the field of fully nonlinear elliptic equations has recently attracted a great deal of attention see e g a recent paper of g m lieberman and n s trudinger 6 and the references therein to other work of the same authors as well as that of l evans n krylov p l lions and others it appears that the case of second order boundary operators is not well studied particularly for noncoercive problems we present an existence result for l which extends the one in 3 where we had considered the equation

asymptotics of block toeplitz determinants and the classical dimer model we compute the asymptotics of a block toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer monomer correlation function the model depends on a parameter interpolating between the square lattice t 0 and the triangular lattice t 1 and we obtain the asymptotics for 0 t 1 for 0 t 1 we apply the szego limit theorem for block toeplitz determinants the main difficulty is to evaluate the constant term in the asymptotics which is generally given only in a rather abstract form

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

two dimensional nonlinear boundary value problems for elliptic equations on considere la regularite limite des solutions du probleme aux valeurs limites non lineaire f x u du d 2 u 0 dans g x u du 0 sur pour des domaines a 2 dimensions f est uniformement elliptique et g depend de du on demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de f g et

the nonlinear oblique derivative problem for quasilinear elliptic equations on considere des problemes aux valeurs limites pour une equation elliptique quasilineaire d ordre 2 qu 0 dans ou est un domaine borne de r n n 2 et q est donne sur c 2 par qu x a ij x u x du x d ij u x a x u x du x

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

a criterion on the hyponormality of toeplitz operators with polynomial symbols via schur numbers in this paper we give a complete characterization on the hyponormality of the toeplitz operators t with trigonometric polynomial symbols g h g h h2 when g divides h this is accomplished by using a criterion on the contractivity of the inverse of a lower triangular finite toeplitz matrix via schur numbers of an induced analytic polynomial

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

covering theorems inequalities on metric spaces and applications to pde s we establish a covering lemma of besicovitch type for metric balls in the setting of holder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets for families of measurable functions we introduce the notions of power decay critical density and double ball property and with the aid of the covering theorem we show how these notions are related next we present an axiomatic procedure to establish harnack inequality that permits to handle both divergence and non divergence linear equations

regularity for fully nonlinear elliptic equations with oblique boundary conditions in this paper we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions in particular we derive the pointwise c c 1 and c 2 regularity as byproducts we also prove the a b p maximum principle harnack inequality uniqueness and solvability of the equations

the viscosity method for the homogenization of soft inclusions in this paper we consider periodic soft inclusions te with periodicity e where the solution ue satisfies semi linear elliptic equations of non divergence in omega epsilon omega setminus overline t epsilon with neumann data on partial t mathfrak a the difficulty lies in the non divergence structure of the operator where the standard energy method which is based on the divergence theorem cannot be applied the main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of ue referred to as u as e approaches to zero we introduce the concept of a compatibility condition between the equation and the neumann condition on the boundary for the existence of uniformly bounded periodic first correctors the concept of a second corrector is then developed to show that the limit u is the viscosity solution of a homogenized equation

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

normal singular cauchy integral operators with operator valued symbols abstract in this paper we characterize a class of normal isometric coisometric unitary hyponormal singular cauchy integral operators with operator valued or matrix valued symbols on vector valued or c n valued l 2 spaces

hyponormality of toeplitz operators with polynomial symbols a bounded linear operator a on a hilbert space h with inner product is said to be hyponormal if its selfcommutator a a a a aa induces a positive semidefinite quadratic form on h via 7 a a for h let h t denote the hardy space of the unit circle t d in the complex plane recall that given l t the toeplitz operator with symbol is the operator t on h t defined by t f p f where f h t and p denotes the projection that maps l t onto h t the hyponormality of toeplitz operators has been studied by c cowen 1 2 p fan 4 c gu 8 t ito and t wong 9 t nakazi and k takahashi 11 d yu 13 k zhu 14 r curto d farenick the second and the third named authors 3 5 6 10 and others an elegant theorem of c cowen 2 characterizes the hyponormality of a toeplitz operator t on h t by properties of the symbol l t k zhu 14 reformulated cowen s criterion and then showed that the hyponormality of t with polynomial symbols can be decided by a method based on the classical interpolation theorem of i schur 12 also farenick and the third named author 5 characterized the hyponormality of t in terms of the fourier coefficients of the trigonometric polynomial in the cases that the outer coefficients of have the same modulus but the case of arbitrary trigonometric polynomials though solved in principle by cowen s theorem or zhu s theorem is in practice very complicated on the other hand nakazi and takahashi 11 corollary 5 showed that if z n n m anz n is a trigonometric polynomial with m n and if for every zero of z such that 1 the number 1 is a zero of z in the open unit disk d of multiplicity greater than or equal to the multiplicity of then t is hyponormal but the converse is not true in general to see this consider the following trigonometric polynomial z z 2 z 2 z 1 z 15 z 13 then z 2 15z 2 19 15z 55 15 53 15z z using an argument of p fan 4 theorem 1 for every trigonometric polynomial of the form z 2 n 2 anz

hyponormality of toeplitz operators with rational symbols in this article we introduce a notion of division for rational functions and then give a criterion for hyponormality of open image in new window f g are rational functions in the cases where g divides f furthermore we show that we may assume without loss of generality that g divides f when we consider the hyponormality of open image in new window

conditions for bound states in a periodic linear chain and the spectra of a class of toeplitz operators in terms of polylogarithm functions conditions for bound states for a periodic linear chain are given within the framework of an exactly solvable non relativistic quantum mechanical model in three dimensional space these conditions express the strength parameter in terms of the distance between two consecutive centres of the chain and of the range interaction parameter this expression can be formulated in terms of polylogarithm functions and in some particular cases in terms of the riemann zeta function an interesting mathematical result is that these expressions also correspond to the spectra of toeplitz complex symmetric operators the non trivial zeros of the riemann zeta function are interpreted as multiple points at the origin of the spectra of these toeplitz operators

subnormal operators regarded as generalized observables and compound system type normal extension related to 1 1 in this paper subnormal operators not necessarily bounded are discussed as generalized observables in order to describe not only the information about the probability distribution of the output data of their measurement but also the framework of their implementations we introduce a new concept of a compound system type normal extension and we derive the compound system type normal extension of a subnormal operator which is defined from an irreducible unitary representation of the algebra 1 1 the squeezed states are characterized as the eigenvectors of an operator from this viewpoint and the squeezed states in multi particle systems are shown to be the eigenvectors of the adjoints of these subnormal operators under a representation the affine coherent states are discussed in the same context as well

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

global w 2 p estimates for solutions to the linearized monge ampere equations in this paper we establish global w 2 p estimates for solutions to the linearized monge ampere equations under natural assumptions on the domain monge ampere measures and boundary data our estimates are affine invariant analogues of the global w 2 p estimates of winter for fully nonlinear uniformly elliptic equations and also linearized counterparts of savin s global w 2 p estimates for the monge ampere equations

solvability of quasilinear elliptic equations with nonlinear boundary conditions ii abstract it is shown that solvability of the second order quasilinear elliptic equation qu 0 in with first order nonlinear boundary condition nu 0 on can be inferred from appropriate estimates on solutions of the problem qu in nu 0 on as varies over a suitable function class this result improves previous work of the author where estimates were required on solutions of qu in nu on as varies over some function space the value of this improvement is demonstrated by some examples

a generalization of cowen s characterization of hyponormal toeplitz operators abstract in this paper we extend an elegant result of c c cowen proc amer math soc 103 1988 809 812 on the hyponormality of toeplitz operators to a class of generalized toeplitz operators in order to do this by a commutant lifting approach we study the operator equation h a h b t c where h a h b are generalized hankel operators and t c is an analytic toeplitz operator related results about the range inclusion of hankel and toeplitz operators are also obtained

the yamabe problem for higher order curvatures let m be a compact riemannian manifold of dimension n 2 the k curvature for k 1 2 n is defined as the k th elementary symmetric polynomial of the eigenvalues of the schouten tenser the k yamabe problem is to prove the existence of a conformal metric whose k curvature is a constant when k 1 it reduces to the well known yamabe problem under the assumption that the metric is admissible the existence of solutions is known for the case k 2 n 4 for locally conformally flat manifolds and for the cases k n 2 in this paper we prove the solvability of the k yamabe problem in the remaining cases k n 2 under the hypothesis that the problem is variational this includes all of the cases k 2 as well as the locally conformally flat case

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

boundary regularity for solutions to the linearized monge ampere equations we obtain boundary holder gradient estimates and regularity for solutions to the linearized monge ampere equations under natural assumptions on the domain monge ampere measures and boundary data our results are affine invariant analogues of the boundary holder gradient estimates of krylov

gradient bounds for solutions of nonuniformly elliptic oblique derivative problems introduction in the theory of quasilinear elliptic equations there are three archetypical forms which seem to embody the various possible behaviors of solutions laplace s equation or more generally a uniformly elliptic equation the minimal surface equation and a sort of false minimal surface equation sii diudiu d u 0 for each of these equations the behavior of solutions to the dirichlet problem is wellunderstood see for example 3 5 10 11 additionally uniformly elliptic conormal problems were studied in 5 chapter lo and minimal surface type conormal problems xvere studied in l 13 61 the class of equations considered in 6 was much broader than those in l 131 but did not include the false minimal surface equation finally uniformly elliptic oblique derivative problems were studied in 7 91 while minimal surface type oblique derivative problems are the subject of 4 here we examine the oblique derivative problem on a smooth domain q c 2

hyponormality and subnormality of block toeplitz operators abstract in this paper we are concerned with hyponormality and subnormality of block toeplitz operators acting on the vector valued hardy space h c n 2 of the unit circle first we establish a tractable and explicit criterion on the hyponormality of block toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator second we consider the gap between hyponormality and subnormality for block toeplitz operators this is closely related to halmos s problem 5 is every subnormal toeplitz operator either normal or analytic we show that if is a matrix valued rational function whose co analytic part has a coprime factorization then every hyponormal toeplitz operator t whose square is also hyponormal must be either normal or analytic third using the subnormal theory of block toeplitz operators we give an answer to the following toeplitz completion problem find the unspecified toeplitz entries of the partial block toeplitz matrix a u u so that a becomes subnormal where u is the unilateral shift on h 2

hyponormality of trigonometric toeplitz operators in this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric toeplitz operators i e toeplitz operators t with trigonometric polynomial symbols our criterion involves the zeros of an analytic polynomial f induced by the fourier coefficients of moreover the rank of the selfcommutator of t is computed from the number of zeros of f in the open unit disk d and in c d counting multiplicity

block toeplitz operators with rational symbols ii in this paper we derive a formula for the rank of the self commutator of hyponormal block toeplitz operators t with matrix valued rational symbols in via the classical hermite fej r interpolation problem

global regularity of solutions of nonlinear second order elliptic and parabolic differential equations on considere des solutions du probleme de dirichlet pour une equation elliptique totalement non lineaire dans un domaine borne lisse et du probleme parabolique analogue

kernels of hankel operators and hyponormality of toeplitz operators we give a formula for mathrm ker h overline theta 1 ast h overline theta 2 and describe when mathrm ker h overline theta 1 ast h overline theta 2 mathrm ker h overline theta 2 we explore the hyponormality of toeplitz operators whose symbols are of circulant type and some more general types in addition we discuss formulas for and estimates of the rank of the self commutator of a hyponormal toeplitz operator

boundary regularity for the monge ampere and affine maximal surface equations in this paper we prove global second derivative estimates for solutions of the dirichlet problem for the monge ampere equation when the inhomoge neous term is only assumed to be holder continuous as a consequence of our approach we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

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

regularity of solutions of nonlinear elliptic boundary value problems on utilise des estimations de type schauder pour ameliorer la regularite supposee des solutions de problemes aux limites non lineaires elliptiques

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

the neumann problem for equations of monge ampere type we prove the existence of a classical solution to a neumann type problem for nonlinear elliptic equations of monge ampere type the methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed gauss curvature we also present applications to asymptotic problems which approach the non unique case and maximum principles for linear equations of aleksandrov bakelman type

interior gradient estimates for solutions to the linearized monge ampere equation let be a convex function on a convex domain rn n 1 the corresponding linearized monge ampere equation is trace d2u f where detd2 d2 1 is the matrix of cofactors of d2 we establish interior holder estimates for derivatives of solutions to such equation when the function f on the right hand side belongs to lp for some p n the function is assumed to be such that c with 0 on and the monge ampere measure detd2 is given by a density g c which is bounded away from zero and infinity

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions

interior second derivative estimates for solutions to the linearized monge ampere equation let rn be a bounded convex domain and c be a convex function such that is sufficiently smooth on and the monge ampere measure det d2 is bounded away from zero and infinity in the corresponding linearized monge ampere equation is trace d2u f where det d2 d2 1 is the matrix of cofactors of d2 we prove a conjecture in gt about the relationship between lp estimates for d2u and the closeness between det d2 and one as a consequence we obtain interior w2 p estimates for solutions to such equation whenever the measure det d2 is given by a continuous density and the function f belongs to lq for some q max p n