diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

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

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

a note on the koekoek s differential equation for generalized jacobi polynomials abstract in a recent paper differential equations for generalized jacobi polynomials submitted for publication koekoek and koekoek discovered a linear differential equation for the polynomials pn m n x n 0 which are orthogonal on 1 1 with respect to 0 1 2 2 1 1 1 1 x 1 x m x 1 n x 1 1 m n 0 this differential equation is of infinite order except in a number of cases it is the purpose of this note to reprove and interpret the results of the koekoeks in the finite order cases in a short and easy way

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

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 uniqueness analysis of a detached shock problem for the potential flow abstract we study a problem for two dimensional steady potential and isentropic euler equations in a bounded domain where an artificial detached shock interacts with a wedge using the stream function we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted holder spaces the proof is based on various holder and schauder estimates for second order elliptic equations and fixed point theorems moreover we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional

ecoulements de fluides parfaits en deux variables independantes de type espace reflexion d un choc plan par un diedre compressif we consider the euler equations of a perfect fluid having only two independent space like variables which account for the stationary 2 dimensional or axisymmetrical 3 dimensional cases as well as the 2 dimensional riemann problem we show that the pressure and the angle between an axis and the velocity field satisfy a first order system which turns out to be elliptic in the subsonic zone in particular the pressure satisfies a maximum principle which has not been stated before to the best of my knowledge using this and the bernouilli law we give various a priori estimates of the pressure the density the enthalpy and the velocity in the problem of the reflection of a shock wave by a wedge we also bound the size of the subsonic region and the force that the fluid applies to the boundary

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

differential equations for generalized jacobi polynomials abstract we look for differential equations of the form m i 0 a i x y i x n i 0 b i x y i x mn i 0 c i x y i x 1 x 2 y x 2 x y x n n 1 y x 0 satisfied by the generalized jacobi polynomials pn m n x n 0 which are orthogonal on the interval 1 1 with respect to the weight function 2 2 1 1 1 1 x 1 x m x 1 n x 1 where 1 1 m 0 and n 0 we give explicit representations for the coefficients ai x i 0 bi x i 0 and ci x i 0 and we show that this differential equation is uniquely determined for m2 n2 0 the order of this differential equation is infinite except for 0 1 2 or 0 1 2 moreover the order equals 2 4 if m 0 n 0 and 0 1 2 2 4 if m 0 n 0 and 0 1 2 2 2 6 if m 0 n 0 and 0 1 2

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

the structure of solutions near a sonic line in gas dynamics via the pressure gradient equation abstract a key issue in gas dynamics in two space dimensions is the regularity of solutions near a sonic curve we build a large class of regular solutions with given boundary conditions on the sonic line the modeling equation is the pressure gradient equation which is the same as euler system when the parameters of the gas are pushed to certain extreme we use a novel set of coordinates involving both the space time and state variables to split regular terms from singular terms in the analysis

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

on the revision of probabilistic beliefs using uncertain evidence we revisit the problem of revising probabilistic beliefs using uncertain evidence and report results on several major issues relating to this problem how should one specify uncertain evidence how should one revise a probability distribution how should one interpret informal evidential statements should and do iterated belief revisions commute and what guarantees can be offered on the amount of belief change induced by a particular revision our discussion is focused on two main methods for probabilistic revision jeffrey s rule of probability kinematics and pearl s method of virtual evidence where we analyze and unify these methods from the perspective of the questions posed above

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

singular second order operators the maximal and minimal operators and selfadjoint operators in between differential operators arising from the differential equation py qy lambda wy are put in the hilbert space setting l 2 a b w a new and simpler characterization of the minimal operator is exhibited selfadjoint operators which lie in between the minimal and maximal operator are easily described in terms of conditions on boundary coefficients which look like and indeed are the same as those imposed on regular problems examples drawn from mathematical physics include the legendre laguerre hermite and bessel problems

some functions that generalize the krall laguerre polynomials let l alpha be the semi infinite tridiagonal matrix associated with the three term recursion relation satisfied by the laguerre polynomials with weight function 1 gamma alpha 1 z alpha e z alpha 1 on the interval 0 infinity we show that when alpha is a positive integer by performing at most alpha successive darboux transformations from l alpha we obtain orthogonal polynomials on 0 infinity with weight distribution 1 gamma alpha k 1 z alpha k e z sigma j 1 k s j delta k j z with 1 less than or equal to k less than or equal to alpha we prove that as a consequence of the rational character of the darboux factorization these polynomials are eigenfunctions of a finite order differential operator our construction calls for a natural bi infinite extension of these results with polynomials replaced by functions of which the semi infinite case is a limiting situation c 1999 elsevier science b v all rights reserved

shock diffraction by convex cornered wedges for the nonlinear wave system we are concerned with rigorous mathematical analysis of shock diffraction by two dimensional convex cornered wedges in compressible fluid flow through the nonlinear wave system this shock diffraction problem can be formulated as a boundary value problem for second order nonlinear partial differential equations of mixed elliptic hyperbolic type in an unbounded domain it can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition we establish a global theory of existence and optimal regularity for this shock diffraction problem to achieve this we develop several mathematical ideas and techniques which are also useful for other related problems involving similar analytical difficulties

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

differential operators having symmetric orthogonal polynomials as eigenfunctions abstract let the polynomials pn x n 0 orthogonal with respect to a symmetric positive definite moment functional be eigenfunctions of a linear differential operator l we consider the orthogonal polynomials pn x n 0 and pn v x n 0 which are obtained by adding one resp two symmetric sobolev type terms to in all the cases we derive a representation for the polynomials and show that they are eigenfunctions of one or more linear differential operators mostly of infinite order of the form l a resp l a v b v c further it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues

non existence of strong regular reflections in self similar potential flow abstract we consider shock reflection which has a well known local non uniqueness the reflected shock can be either of two choices called weak and strong we consider cases where existence of a global solution with weak reflected shock has been proven for compressible potential flow if there was a global strong shock solution as well then potential flow would be ill posed however we prove non existence of strong shock analogues in a natural class of candidates

on boundary values for the laguerre operator in indefinite inner product spaces abstract it is shown that when ly xy 1 x y requires more than one boundary condition at x 0 in order to be self adjoint the more negative the more boundary conditions required

on markov s theorem on zeros of orthogonal polynomials revisited abstract this paper briefly surveys markov s theorem related to zeros of orthogonal polynomials monotonicity of zeros of some families of orthogonal polynomials are reviewed in detail

differential equations having orthogonal polynomial solutions abstract necessary and sufficient conditions for an orthogonal polynomial system ops to satisfy a differential equation with polynomial coefficients of the form l n y i 1 n l i x y i x n y x were found by h l krall here we find new necessary conditions for the equation to have an ops of solutions as well as some other interesting applications in particular we obtain necessary and sufficient conditions for a distribution w x to be an orthogonalizing weight for such an ops and investigate the structure of w x we also show that if the equation has an ops of solutions which is orthogonal relative to a distribution w x then the differential operator ln in must be symmetrizable under certain conditions on w x

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

constraints on possible singularities for the unsteady transonic small disturbance utsd equations we discuss the singular behavior of solutions to two dimensional general secondorder uniformly elliptic equations in divergence form with bounded measurable coefficients and discontinuous dirichlet data along a portion of a lipschitz boundary we show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data a problem like this arises in the study of self similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance utsd flow equations these solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von neumann paradox applies the present result provides a step in the direction of understanding the nature of the solutions to the utsd equations near a triple point it shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions c 1999 john wiley sons inc

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations we consider solutions and subsolutions and supersolutions of the boundary value problem au x u du diju u x u du 0 in 2 s x diu y x u g x on3fi for a lipschitz domain s a positive definite matrix valued function u j and a vector field s which points uniformly into q without making any continuity assumptions on the known functions we prove harnack and holder estimates for u near 3i2 in addition we bound the l norm of u near 3i2 in terms of an appropriate lp norm and the known functions our approach is based on that for the corresponding interior estimates of trudinger this paper is concerned with analogs for solutions of oblique boundary value problems of the results in 10 our results will extend previous work of lieberman and trudinger 5 and of nadirashvili 9 specifically we consider second order differential operators q of the form

nonexistence of the von neumann reflection configuration for the triple point paradox in an attempt to resolve the von neumann triple point paradox a reflection called von neumann reflection vnr was reported by colella and henderson j fluid mech 213 1990 pp 71 94 in investigating numerically the weak shock reflection they reported that in this type of reflection configuration the incident shock and the mach stem are seen to be smoothly merged and the reflected disturbance is not a shock but a curved band of self similar compression wave one should like to know whether or not the vnr configuration is a mathematically possible flow pattern in this paper we prove that the reflection configuration in which the mach shock is smoothly merged into the incident shock at a point and the wave behind this point is smooth is a mathematically impossible flow pattern for the two dimensional 2d self similar potential flow equation and the 2d self similar euler equations

counterexamples to the sonic criterion we consider self similar pseudo steady shock reflection at an oblique wall there are three parameters wall corner angle mach number angle of incident shock ever since ernst mach discovered the irregular reflection named after him researchers have sought to predict precisely for which parameters the reflection is regular three conflicting proposals the detachment sonic and von neumann criteria have been studied extensively without a clear result we demonstrate that the sonic criterion is not correct we consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is transonic we also provide a clear physical reason the flow type sub or supersonic is not decisive instead the reflected shock type weak or strong determines whether structural perturbations decay towards the reflection point

interaction of four rarefaction waves in the bi symmetric class of the two dimensional euler equations the global existence and structures of solutions to multi dimensional unsteady compressible euler equations are interesting and important open problems in this paper we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the euler equations in two space dimensions in the case where the initial rarefaction waves are large the bi symmetric initial data is a basic type of four wave two dimensional riemann problems the solutions in this case are continuous bounded and self similar and we characterize how large the rarefaction waves must be we use the methods of hodograph transformation characteristic decomposition and phase space analysis we resolve binary interactions of simple waves in the process

construction of differential operators having bochner krall orthogonal polynomials as eigenfunctions abstract suppose q n n 0 is a sequence of polynomials orthogonal with respect to the moment functional where is a classical moment functional jacobi laguerre hermite and is a point mass distribution with finite support in this paper we develop a new method for constructing a differential equation having q n n 0 as eigenfunctions

left definite theory with applications to orthogonal polynomials in the past several years there has been considerable progress made on a general left definite theory associated with a self adjoint operator a that is bounded below in a hilbert space h the term left definite has its origins in differential equations but littlejohn and wellman l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 generalized the main ideas to a general abstract setting in particular it is known that such an operator a generates a continuum h r r 0 of hilbert spaces and a continuum of a r r 0 of self adjoint operators in this paper we review the main theoretical results in l l littlejohn r wellman a general left definite theory for certain self adjoint operators with applications to differential equations j differential equations 181 2 2002 280 339 moreover we apply these results to several specific examples including the classical orthogonal polynomials of laguerre hermite and jacobi

djkm algebras and non classical orthogonal polynomials abstract we describe families of polynomials arising in the study of the universal central extensions of lie algebras introduced by date jimbo kashiwara and miwa 1983 6 in their work on the landau lifshitz equations we show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials

on a generalization of bessel functions satisfying higher order differential equations bessel type functions j m x 0 with two parameters 12 and m 0 which include the classical normalized bessel function for m 0 are introduced as a certain confluent limit of koornwinder s laguerre type polynomials l nn x n n0 for any n0 they arise as solutions of a spectral dependent differential equation of order 2 4 a necessary and sufficient condition to transform the differential equation into symmetric form is given in terms of an overdetermined system of linear equations it is shown that for 0 1 2 a solution of this problem exists and leads to a symmetric fourth sixth and eighth order differential equation respectively for any 12 we also derive a second order differential equation but with coefficients depending nonlinearly on the eigenvalue parameter finally two differential expressions of order 2 2 n0 are constructed which map the classical bessel functions onto their nonclassical counterparts and vice versa this result may be used to establish an orthogonality relation for the bessel type functions in a distributional sense with respect to the distributional weight function 2 1 x2 1 m x denoting the point mass centered at the origin

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

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

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result