estimates for the maximal singular integral in terms of the singular integral the case of even kernels let t be a smooth homogeneous calder on zygmund singular integral operator in r n in this paper we study the problem of controlling the maximal singular integral t f by the singular integral tf the most basic form of control one may consider is the estimate of the l 2 r n norm of t f by a constant times the l 2 r n norm of tf we show that if t is an even higher order riesz transform then one has the stronger pointwise inequality t f x c m tf x where c is a constant and m is the hardy littlewood maximal operator we prove that the l 2 estimate of t by t is equivalent for even smooth homogeneous calder on zygmund operators to the pointwise inequality between t and m t our main result characterizes the l 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel x jxjn of t where is an

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

remarks on the repulsive wigner poisson system we present an estimate for the space time integral of classical solutions to the repulsive wigner poisson system we use markowich s formalism between wigner possion and schrodinger poisson systems through this formalism and morawetz interaction potentials we derive the same a priori estimate given by chae and ha for the repulsive vlasov poisson system

on the periodic wigner poisson fokker planck system abstract this paper is concerned with the existence and uniqueness analysis of global classical solutions of a diffusive quantum evolution equation with nonlinear coupling to the poisson equation the main technical difficulty in the existence proof is to show that the quantum fokker planck term is a semigroup generator in a weighted l 2 space the potential term is then a lipschitz perturbation of it

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

an operator splitting method for the wigner poisson problem the wigner poisson equation describes the quantum mechanical motion of electrons in a self consistent electrostatic field the equation consists of a transport term and a non linear pseudodifferential operator in this paper we analyze an operator splitting method for the linear wigner equation and the coupled wigner poisson problem for this semidiscretization in time consistency and nonlinear stability are established in an l2 framework we present a numerical example to illustrate the method

analytic functions sharing level curves and tracts a level curve for a nonconstant analytic function is a curve on which the function has constant modulus analytic functions should be symmetric about their level curves any behavior on one side should be the reflection of behavior on the other this symmetry may be realized locally one straightens out a bit of the level curve with a local change of coordinates and appeals to the schwarz reflection principle here we prove that in fact the only obstructions to global symmetry are presented by compositional factors removing these factors allows us to straighten out complete level curves and to exploit the perfect symmetry of the functions remaining we can then establish in a unified way numerous old and new results about shared level curves and tracts including work going back to valiron and cartwright and more recent work of fuchs heins and others given a riemann surface o and a nontrivial arc r in 4 we denote by yr the nonconstant meromorphic functions having constant modulus one on f when 4 is the riemann sphere p the plane c or the unit disc d and f lies in the extended real line rx we use the notation 3vr the key theorem of our work is this

evaluation representations of quantum affine algebras at roots of unity the purpose of this paper is to compute the drinfel d polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation schnizer modules moreover we obtain the necessary and sufficient condition for that the two types of evaluation representations are isomorphic to each other

qualitative behavior and traveling waves for flux saturated porous media equations arising in optimal mass transportation abstract this paper is devoted to the analysis of qualitative properties of porous media flux saturated type operators which combine the effects of reaction terms of fkpp type in dimension one specifically we study the existence of traveling wave profiles sub and super solutions waiting time and dynamical properties of the support the paper also deals with the connection between these minimal traveling waves with those of other related models porous media flux saturated mechanism and the heat equation

the weak hopf algebras related to generalized kac moody algebra we define a kind of quantized enveloping algebra of a generalized kac moody algebra g by adding a generator j satisfying jm jm 1 for some integer m we denote this algebra by wuq g this algebra is a weak hopf algebra if and only if m 2 in general it is a bialgebra and contains a hopf subalgebra this hopf subalgebra is isomorphic to the usually quantum envelope algebra uq g of a generalized kac moody algebra g

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

an analysis of the wigner poisson problem with inflow boundary conditions abstract we present a study of the wigner poisson problem in a bounded spatial domain with non homogeneous and time dependent inflow boundary conditions this system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts we prove well posedness of the linearized n dimensional problem as well as existence and uniqueness of a global in time regular solution of the one dimensional nonlinear problem

traveling waves in a reaction diffusion system diffusion with finite velocity and kolmogorov petrovskii piskunov kinetics an asymptotic method is presented for the analysis of the traveling waves in the one dimensional reaction diffusion system with the diffusion with a finite velocity and kolmogorov petrovskii piskunov kinetics the analysis makes use of the path integral approach scaling procedure and the singular perturbation techniques involving the large deviations theory for the poisson random walk the exact formula for the position and speed of reaction front is derived it is found that the reaction front dynamics is formally associated with the relativistic hamiltonian lagrangian mechanics

pattern formation in a flux limited reaction diffusion equation of porous media type a non linear pde featuring flux limitation effects together with those of the porous media equation non linear fokker planck is presented in this paper we analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns namely traveling wave solutions to this singular reaction diffusion equation we show the existence and qualitative behavior of different types of traveling waves classical profiles for wave speeds high enough and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold some of these solutions are of particular relevance as they provide models by which the whole solution and not just the bulk of it as it is the case with classical traveling waves spreads through the medium with finite speed

analysis of a semidiscrete version of the wigner equation we introduce a semidiscretized version of the wigner equation discretization concerning the velocity variable we show that the corresponding discrete velocity problem is well posed and permits us to approach the solution of the continuous problem when the mesh size of the discretization vanishes the approximation shows spectral accuracy because the rate of convergence corresponds to the sobolev regularity of the solution of the continuous problem we also discuss the behavior of the solution with respect to the planck constant

time dependent dissipation in nonlinear schrodinger systems a coupled nonlinear schrodinger poisson equation is considered which contains a time dependent dissipation function as a specific model of dissipation effects in nonlinear quantum transport theory and other areas the wigner poisson equation associated with this system is derived using conservation and quasiconservation laws and certain growth assumptions for the nonlinearities and the dissipation function global existence of solutions to the cauchy problem of the time dependent schrodinger poisson system is shown both for small attractive case or arbitrary data repulsive case

a strongly degenerate quasilinear elliptic equation abstract we prove existence and uniqueness of entropy solutions for the quasilinear elliptic equation u div a u du v where 0 v l 1 r n l r n a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class of equations includes the elliptic problems associated to a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics respectively in a second part of this work using crandall liggett s iteration scheme this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic cauchy problem

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

a martin boundary in the plane let e be an open connected subset of euclidean space with a green function and let a be harmonic measure on the martin boundary i of e we will show that except for a a x a null set of x y g i 2 x is an entrance point for brownian motion conditioned to leave e at y r s martin gave examples in dimension 3 or higher for which there exist minimal accessible martin boundary points x 7 y for which this condition fails we will give a similar example in dimension 2

sign and area in nodal geometry of laplace eigenfunctions the paper deals with asymptotic nodal geometry for the laplace beltrami operator on closed surfaces given an eigenfunction f corresponding to a large eigenvalue we study local asymmetry of the distribution of sign f with respect to the surface area it is measured as follows take any disc centered at the nodal line f 0 and pick at random a point in this disc what is the probability that the function assumes a positive value at the chosen point we show that this quantity may decay logarithmically as the eigenvalue goes to infinity but never faster than that in other words only a mild local asymmetry may appear the proof combines methods due to donnelly fefferman and nadirashvili with a new result on harmonic functions in the unit disc

existence and uniqueness of a solution for a parabolic quasilinear problem for linear growth functionals with l 1 data we introduce a new concept of solution for the dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for these solutions to prove the existence we use the nonlinear semigroup theory

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

stationary wigner equation with inflow boundary conditions will a symmetric potential yield a symmetric solution based on the well posedness of the stationary wigner equation with inflow boundary conditions given in a arnold h lange and p f zweifel j math phys 41 2000 pp 7167 7180 we prove without any additional prerequisite conditions that the solution of the wigner equation with inflow boundary conditions will be symmetric only if the potential is symmetric this improves the result in d taj l genovese and f rossi europhys lett 74 2006 pp 1060 1066 which depends on the convergence of the solution formulated in the neumann series by numerical studies we present the convergence of the numerical solution to the symmetric profile for three different numerical schemes this implies that the upwind schemes can also yield a symmetric numerical solution contrary to the argument given in d taj l genovese and f rossi europhys lett 74 2006 pp 1060 1066

a faber krahn inequality for solutions of schrodinger s equation abstract we consider nontrivial solutions of u x v x u x where u 0 on the boundary of a bounded open region d r n and v x l d we prove a sharp relationship between v and the measure of d which generalizes the well known faber krahn theorem we also prove some geometric properties of the zero sets of the solution of the schrodinger equation u x v x u x

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

a class of superintegrable systems of calogero type we show that the three body calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in euclidean three space as such it is a special case of a family of systems involving one arbitrary function of one variable

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

finite propagation speed for limited flux diffusion equations we prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed identified as the speed of light c for that we construct entropy sub and super solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub and super solutions respectively this enables us to prove the existence of discontinuity fronts moving at light s speed

on the nodal sets of toral eigenfunctions we study the nodal sets of eigenfunctions of the laplacian on the standard d dimensional flat torus the question we address is can a fixed hypersurface lie on the nodal sets of eigenfunctions with ar bitrarily large eigenvalue in dimension two we show that this happens only for segments of closed geodesics in higher dimensions certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues our main result is that this cannot happen for hypersur faces with nonzero gauss kronecker curvature in dimension two the result follows from a uniform lower bound for the l 2 norm of the restriction of eigenfunctions to the curve proved in an earlier paper 1 in high dimensions we currently do not have this bound instead we make use of the real analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain in all of our results we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere we also present an independent proof for the two dimensional case relying on the abc theorem in function fields

a spectral method for the numerical simulation of quantum tunneling phenomena tunneling effects in solid state physics are frequently described via a quantum mechanical formulation based on the wigner distribution function in this paper a mixed finite difference spectral collocation method for the solution of the corresponding equation of motion is developed and analyzed numerical examples illustrating the convergence results are presented

on the three dimensional wigner poisson problem with inflow boundary conditions abstract we study the wigner poisson problem in a bounded spatial domain with non homogeneous and time dependent inflow boundary conditions this system is a quantum model of charge transport in a semiconductor device coupled with reservoirs in presence of a self consistent potential and of an external one we state a local in time well posedness result for the problem the main difficulty is proving in the three dimensional case that the non linear potential term is a lipschitz perturbation of the affine streaming operator in an appropriately weighted l 2 space

operator splitting methods applied to spectral discretizations of quantum transport equations the wigner poisson equation is the most successful basis for transient simulations of quantum effect semiconductor devices so far we present a full discretization of this nonlinear pseudo differential system using a mixed spectral collocation and operator splitting method convergence and nonlinear stability of the scheme are proven

space time discretization of series expansion methods for the boltzmann transport equation the approximate solution of the boltzmann transport equation via galerkin type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients this system is extremely stiff close to the fluid dynamical regime for small knudsen numbers and exhibits a mildly dispersive behavior due to the acceleration of waves by the external force the electric field in this paper a class of difference methods is presented and analyzed which represent a generalization of the well known scharfetter gummel exponential fitting approach for the drift diffusion equations it is shown that by using appropriate operator splitting methods for the time discretization one obtains stability properties which are only mildly dependent on the knudsen number and essentially independent of the size of the electric field

a fisher kolmogorov equation with finite speed of propagation abstract in this paper we study a fisher kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m

some regularity results on the relativistic heat equation we prove some partial regularity results for the entropy solution u of the so called relativistic heat equation in particular under some assumptions on the initial condition u0 we prove that ut t is a radon measure in rn moreover if u0 is log concave inside its support being a convex set then we show the solution u t is also log concave in its support t this implies its smoothness in t in that case we can give a simpler characterization of the notion of entropy solution

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

on ratios of harmonic functions abstract let u and v be harmonic functions in r n with the same zero set z we show that the ratio f of such functions is always well defined and is real analytic moreover it satisfies the maximum and minimum principles for n 3 we also prove the harnack inequality and the gradient estimate for the ratios of harmonic functions namely sup k f c inf k f sup k f c inf k f for any compact subset k of where the constant c depends on k z only in dimension two the first inequality follows from the boundary harnack principle and the second from the gradient estimate recently obtained by mangoubi it is an open question whether these inequalities remain true in higher dimensions n 4

nonlinear functionals of the boltzmann equation and uniform stability estimates abstract we present nonlinear functionals measuring physical space variation and l 1 distance between two classical solutions for the boltzmann equation with a cut off inverse power potential in the case that initial datum is a small smooth perturbation of vacuum and decays fast enough in the phase space we show that these functionals satisfy stability estimates which lead to bv type estimates and a uniform l 1 stability

optimal profiles in a phase transition model with a saturating flux abstract it is well known that for the allen cahn equation the minimizing transition in an infinite cylinder r is one dimensional and unique up to a translation in the first variable we analyze in this paper the existence and symmetry of optimal profiles for transitions in a similar phase separation model with a saturating flux this amounts to consider transitions in the space of bv functions as we consider the area integral instead of the dirichlet energy to penalize the creation of wild interfaces

numerical simulation of the quantum liouville poisson system abstract we simulate the one dimensional quantum liouville poisson system using the splitting scheme and the accompanying double fourier transformation in x and p space this code is used to study the quantum effects in the one dimensional electrostatic plasma i e the well known nonlinear landau damping and two stream instability problems

the wignerpoisson x system in wiener algebra in this paper we concern over a class of the wignerpoisson x system introduced and developed by bao etal 2003 mauser 2001 and stimming 2005 respectively the model describes the quantum mechanical motion of particles under the influence of the nonlocal hartree potential and a local power term exchange potential the existence and uniqueness of local mild solution to the n dimensional n 1 3 cauchy problems are established on the space of some integrable functions whose inverse fourier transforms are also integrable