vanishing viscosity with short wave long wave interactions for multi d scalar conservation laws we consider a system coupling a multidimensional semilinear schrodinger equation and a multidimensional nonlinear scalar conservation law with viscosity which is motivated by a model of short wave long wave interaction introduced by benney 1977 we prove the global existence and uniqueness of the solution of the cauchy problem for this system we also prove the convergence of the whole sequence of solutions when the viscosity e and the interaction parameter approach zero so that o e1 2 we also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of schrodinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

nonlinear riemann hilbert problem for bordered riemann surfaces let be a bordered riemann surface with genus g and m boundary components let z z be a smooth family of smooth jordan curves in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i which all contain the point 0 in their interior then there exists a holomorphic function f z on smooth up to the boundary with at most 2 g m 1 zeros on such that f z z for every z

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

polynomially convex hulls with piecewise smooth boundaries is the boundary of a compact convex set y z c c m with nonempty interior set y u z x y z alexander and wermer 1 and slodkowski 8 proved zcbd independently that each point p in the polynomial convex hull of m n p d lies in the graph of a bounded analytic disk f z z f 1 z f z f j h d for l j m with image in 3 r alexander and wermer obtained a more precise information in the case when m 1 and each fiber m z is a circle 1 theorems 2 and 3 in this paper we shall assume the existence of an analytic disk f o z z f0 z on d continuous on such thatfo z is an interior point of y z for each z bd then each bounded analytic disk f z z f z in ai which passes through a boundary point of a r lying over d is entirely contained in the boundary o fair and its cluster set f d f d is contained in m theorem 1 1 if m 1 and m is a smooth submanifold of r as above then the hull 3 r has piecewise smooth boundary and is f dled by the images of analytic disks with smooth boundary values contained in m theorem 1 2 we also obtain a uniqueness result for the representing measures corollary 1 3

iterative methods for discrete nonlinear riemann hilbert problems abstract let d denote the complex unit disk t d and let f t r 2 r be a given function the paper presents iterative methods for the computation of functions w u iv which are holomorphic in d and satisfy the boundary relation f t u t v t 0 for all t t it is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle t the proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution error estimates for the iterative solutions are given in the scale of the sobolev spaces wk2

remark on the scattering operator for the cubic nonlinear dirac equation in three space dimensions abstract this paper is concerned with the scattering operator s for the three dimensional dirac equation with a cubic nonlinearity it follows from known results that s is well defined on a neighborhood of 0 in the sobolev space h r 3 c 4 for any 1 in the present paper we prove that for any m n and s max m there exists some neighborhood u of 0 in the weighted sobolev space h s m r 3 c 4 such that s u h s m r 3 c 4

complex interpolation of normed and quasinormed spaces in several dimensions i a variety of complex interpolation methods for families of normed or quasi normed spaces parametrized by points of domains in complex homo geneous spaces parametrized by points of domains in complex homogeneous spaces is developed results on existence continuity uniqueness reiteration and duality for interpolation are proved as well as on interpolation of oper ators a minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem

polynomial hulls and h infty control for a hypoconvex constraint we say that a subset of mathbb c n is hypoconvex if its complement is the union of complex hyperplanes let delta be the closed unit disk in mathbb c gamma partial delta we prove two conjectures of helton and marshall let rho be a smooth function on gamma times mathbb c n whose sublevel sets have compact hypoconvex fibers over gamma then with some restrictions on rho if y is the set where rho is less than or equal to 1 the polynomial convex hull of y is the union of graphs of analytic vector valued functions with boundary in y furthermore we show that the infimum inf f in h infty delta n rho z f z infty is attained by a unique bounded analytic f which in fact is also smooth on gamma we also prove that if rho varies smoothly with respect to a parameter so does the unique f just found

congruences on regular semigroups let s be a regular semigroup and let p be a congruence relation on s the kernel of p in notation ker p is the union of the idempotent eclasses the trace of p in notation trp is the restriction of p to the set of idempotents of s the pair kerp trp is said to be the congruence pair associated with p congruence pairs can be characterized abstractly and it turns out that a congruence is uniquely determined by its associated congruence pair the triple p v ker p p v r r is said to be the congruence triple associated with p congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple the consideration of the parameters which appear in the above mentioned representations of congruence relations gives insight into the structure of the congruence lattice of s for congruence relations p and 0 put ptz 0 ptrr 0 pto if andonlyifpv 0v pvr 0vr trp tr0 thentz tr and t are complete congruences on the congruence lattice of s and t tl n t introduction and summary after it was realized by wagner that a congruence on an inverse semigroup s is uniquely determined by its idempotent classes preston provided an abstract characterization of such a family of subsets of s called the kernel normal system see 2 chapter 10 this approach was the only usable means for handling congruences on inverse semigroups for two decades a new approach to the problem of describing congruences on inverse semigroups was sparked by the work of scheiblich 13 who described congruences in terms of kernels and traces a systematic exposition of the achievements of this approach can be found in 10 chapter iii it was recognized by feigenbaum 3 that every congruence p on a regular semigroup s is uniquely determined by its kernel kerp equal to elements tequivalent to idempotents and its trace trp equal to the restriction of p to the set e s of idempotents of s in the case of an inverse semigroup s kerp and trp as well as their mutual relationship can be described abstractly by means of simple conditions on a subset of s and an equivalence on e s see 10 chapter iii following in the footsteps of scheiblich for orthordox and arbitrary regular semigroups feigenbaum 3 and lyotter 14 adopted the following approach trp is characterized abstractly and to each such trp all matching kernels are described this unbalances the symmetry of the kernel trace approach by giving preference to the trace hence a balanced view relative to the kernel and the trace is evidently called for the unqualified success of the kernel trace approach for inverse semigroups including its diverse ramifications gave a certain hope that this may also turn out to be the case for regular semigroups judging by the complexity of regular semigroups and the attempts made for both orthodox and general regular semigroups received by the editors november 2 1984 1980 mathematics subject classification primary 20m10 secondary 08a30 a 1986 american mathematical society 0002 9947 86 1 00 25 per page

energy scattering for nonlinear klein gordon and schrodinger equations in spatial dimensions 1 and 2 we show that when n 1 and 2 the scattering operators are well defined in the whole energy space for nonlinear klein gordon and schrodinger equations in r1 n with nonlinearity u p 1 u p 1 4 n such results have been known only for n 3

a theory of complex interpolation for families of banach spaces abstract a detailed development is given of a theory of complex interpolation for families of banach spaces which extends the well known theory for pairs of spaces

extremal discs and the regularity of cr mappings in higher codimension we develop a local theory of extremal analytic discs attached to a strictly pseudoconvex manifold of higher codimension in complex space we then apply the results to prove the regularity of cr homeomorphisms of such manifolds introduction in his celebrated paper lempert li introduced extremal analytic discs as geodesies of the kobayashi metric on a convex domain d c cn let a be the unit disc in the complex plane an analytic disc a d is called extremal if 0 attains its maximum among all analytic discs in d with the same values and directions at 0 g a extremal discs are important holomorphic invariants of d their boundaries lie in the boundary bd of the domain and form an invariant family of curves for the cr structure of bd lempert li l2 applies the extremal discs to the regularity of biholomorphic mappings monge amp re equations and normal forms of convex domains other applications include deformations and embeddibility of cr structures bl bd and polynomial hulls s

on space time means and everywhere defined scattering operators for nonlinear klein gordon equations on demontre l existence d operateurs de diffusion definis partout pour certaines equations de klein gordon non lineaires

scattering problem for a system of nonlinear klein gordon equations related to dirac klein gordon equations abstract we prove the existence of a scattering operator for a system of nonlinear klein gordon equations related to dirac klein gordon equations in three space dimensions

polynomial hulls with convex fibers and complex geodesics let x be a compact subset of z 1 cn with convex fibers several equivalent conditions are obtained that characterize polynomially convex hulls y x with nonempty interior under the latter assumption it is shown that every z0 w0 y and such that z0 1 is contained in a closed n dimensional complex submanifold of d cn which is tangent to y along an analytic disc we show as an application that some results of lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls

on bounded analytic functions in finitely connected domains a new proof of the corona theorem for finitely connected domains is given it is based on a result on the existence of a meromorphic selection from an analytic set valued function the latter fact is also applied to the study of finitely generated ideals of h over multiply connected domains introduction in this paper which is a direct continuation of 18 we study applications of analytic multifunctions to some topics in function theory on finitely connected domains related mainly to the corona problem concerniag analytic multifunctions which are certain set valued functions cf definition 1 2 1 3 of 18 form sufficient background for our purposes here the reader is also referred there for the information about the origin of the approach employed in this paper however basic definitions and references are provided below our basic new result theorem 1 4 associates to every analytic multifunction defined in a finitely connected domain some meromorphic functions with poles at the critical points of green s function of this domain some improvements of this fact in txhe special case of an annulus are discussed in 3 this main technical result is applied in 2 to help obtain a new simple proof of the well known corona theorem for finitely connected domains a novel feature of this proof as compared with e g forelli 9 stout 19 gamelin 7 is that it does not use the corresponding result for the unit disc the same methods are used in 4 where we consider the question when some power of g from h g belongs to the ideal af1 af2 afn we obtain generalizations to multiply connected domains of a result due to t wolff theorems 4 1 and 4 2 as well as its refinements in the case of the unit disc corollary 4 4 and example 5 3 all proofs given in this paper extend to finite riemann surfaces 1 a meromorphic selection theorem the main result of this section relates some meromorphic vector valued functions to analytic multifunctions defined in finitely connected domains first we recall some definitions definition 1 1 folk a locally compact set z c cn is a maximum set if for every compact set n c z and for every analytic function f defined in a neighborhood of n the inequality maxn if l maxazn if l holds definition 1 2 16 18 anuppersemicontinuousset valuedcorrespondence z k z g 2c where g c c is open and all k z are nonempty and received by the editors january 16 1986 presented to the society special session on convex analysis new orleans january 11 1986 1980 mathematics subject classifiratson 1985 ren primary 32a80 32e20 46j15 2 1987 american mathematical society 0002 9947 87 1 00 25 per page

rapidly decaying solutions of the nonlinear schrodinger equation we consider global solutions of the nonlinear schrodinger equation iu t delta u lambda u alpha u in r n nls where r and 0 0 we prove asymptotic completeness for alpha 0 leqq alpha frac 4 n 2

sysygies of unimodular lawrence ideals infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra the quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated binomial ideal which defines a toric subvariety in a product of projective lines connections to graphic arrangements and to beilinson s spectral sequence are explored

high order corrections to the time dependent born oppenheimer approximation ii coulomb systems we study the dynamics of molecular systems with coulomb forces we prove that if the nuclear masses are proportional to e 4 then certain solutions to the time dependent schrodinger equation have asymptotic expansions to arbitrarily high order in powers of e as e 0

on scattering and everywhere defined scattering operators for nonlinear klein gordon equations asymptotic properties of solutions of the nonlinear klein gordon equation t2u u m2u f u 0 nlkg u 0 tu 0 are investigated which are inherited from the corresponding solutions v of the linear klein gordon equation t2v v m2v 0v 0 tv 0 kg in particular the finiteness of time integrals in lq over r of certain sobolevnorms in space of the solution is proved to be such a hereditary property together with a device by w a strauss and a weak decay result for the kg due to r s strichartz this is used to prove that under suitable restrictions on the nonlinearity the scattering operator for the nlkg is defined on all of l21 l2 for n 3

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

low energy scattering for nonlinear klein gordon equations the scattering operator which belongs to a pair of pdes consisting of the klein gordon equation and a perturbation of it by a power like nonlinearity z hfl u is studied it is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if z hfl u u p 1u or u p where 1 4 n 1 p 1 4 n 2 and the space dimension n 2 is arbitrary

scattering operator for nonlinear klein gordon equations in higher space dimensions abstract we prove the existence of the scattering operator in the neighborhood of the origin in the weighted sobolev space h 1 with max 3 2 1 2 n for the nonlinear klein gordon equation with a power nonlinearity u t t u u u 1 u t x r r n where 1 4 n 2 1 4 n for n 3 c