the riemann hilbert problem and the generalized neumann kernel on multiply connected regions this paper presents and studies fredholm integral equations associated with the linear riemann hilbert problems on multiply connected regions with smooth boundary curves the kernel of these integral equations is the generalized neumann kernel the approach is similar to that for simply connected regions see r wegmann a h m murid m m s nasser the riemann hilbert problem and the generalized neumann kernel j comput appl math 182 2005 388 415 there are however several characteristic differences which are mainly due to the fact that the complement of a multiply connected region has a quite different topological structure this implies that there is no longer perfect duality between the interior and exterior problems we investigate the existence and uniqueness of solutions of the integral equations in particular we determine the exact number of linearly independent solutions of the integral equations and their adjoints the latter determine the conditions for solvability an analytic example on a circular annulus and several numerically calculated examples illustrate the results

gevrey regularity of solutions to the non cutoff homogeneous boltzmann equation for soft potential with strong singularity abstract in this study we analyze the cauchy problem for the non cutoff homogeneous boltzmann equation with strong singularity in contrast to our previous work lin 2014 8 the present study considers the soft potential case and derives the gevrey regularity of the c solutions

gelfand shilov smoothing properties of the radially symmetric spatially homogeneous boltzmann equation without angular cutoff we prove that the cauchy problem associated to the radially symmetric spatially homogeneous non cutoff boltzmann equation with maxwellian molecules enjoys the same gelfand shilov regularizing effect as the cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator

spiral slits map and its inverse of bounded multiply connected regions this paper presents a boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto a spiral slits region the method is an extension of the author s method for computing the circular slits map of bounded multiply connected regions see a w k sangawi et al 2012 16 several numerical examples are given to prove the effectiveness of the proposed methods

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

regularizing effect and local existence for the non cutoff boltzmann equation the boltzmann equation without grad s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non integrable angular singularity of the cross section however even though this has been justified satisfactorily for the spatially homogeneous boltzmann equation it is still basically unsolved for the spatially inhomogeneous boltzmann equation in this paper by sharpening the coercivity and upper bound estimates for the collision operator establishing the hypo ellipticity of the boltzmann operator based on a generalized version of the uncertainty principle and analyzing the commutators between the collision operator and some weighted pseudo differential operators we prove the regularizing effect in all time space and velocity variables on the solutions when some mild regularity is imposed on these solutions for completeness we also show that when the initial data has this mild regularity and a maxwellian type decay in the velocity variable there exists a unique local solution with the same regularity so that this solution acquires the c regularity for any positive time

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

a charge simulation method for the numerical conformal mapping of interior exterior and doubly connected domains abstract a simple numerical method is described for computing the following three conformal maps a from a domain interior to a closed jordan curve onto the interior of the unit disk b from a domain exterior to a closed jordan curve onto the exterior of the unit disk and c from a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners

recovering the initial wave amplitude for nonlinear elliptic equation with locally lipschitz source in multiple dimensional domain abstract the backward problems for the elliptic equation bpee are widely used for modelling problems in many fields such as physics geometry or engineering this paper is the first investigation of bpee in unbounded multiple dimensional domain associated with locally lipschitz source as follows 2 u x 1 2 2 u x 2 2 2 u x n 2 2 u y 2 s x 1 x 2 x n y u x 1 x 2 x n y in which x 1 x 2 x n y r n 0 l where l 0 based on the idea of the truncation approach we construct the regularized solution and obtain the h older type of its convergence rate under some assumptions on regularity of the exact solution eventually a numerical experiment for the elliptic allen cahn equation is proposed to illustrate the effectiveness and applicability of our method

numerical conformal mappings of bounded multiply connected domains by the charge simulation method we present here a simple method of numerical conformal mappings from bounded multiply connected domains onto the bounded canonical slit domains of nehari i e a disk with concentric circular slits and an annulus with concentric circular slits in this method we express the mapping functions in terms of a pair of conjugate harmonic functions and approximate them using the charge simulation method by a linear combination of complex logarithmic functions some numerical examples show the effectiveness of the method

ultra analytic effect of cauchy problem for a class of kinetic equations abstract the smoothing effect of the cauchy problem for a class of kinetic equations is studied we firstly consider the spatially homogeneous nonlinear landau equation with maxwellian molecules and inhomogeneous linear fokker planck equation to show the ultra analytic effects of the cauchy problem those smoothing effect results are optimal and similar to heat equation in the second part we study a model of spatially inhomogeneous linear landau equation with maxwellian molecules and show the analytic effect of the cauchy problem

numerical conformal mapping of multiply connected regions onto the fifth category of koebe s canonical slit regions abstract this paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of koebe s classical canonical slit regions the method is based on a uniquely solvable boundary integral equation with generalized neumann kernel the results of some test calculations illustrate the performance of the presented method

gevrey regularity of spatially homogeneous boltzmann equation without cutoff abstract in this paper we study the gevrey regularity of spatially homogeneous boltzmann equation without angular cutoff we prove the propagation of gevrey regularity for c solutions with the maxwellian decay to the cauchy problem of spatially homogeneous boltzmann equation the idea we use here is based on the framework of morimoto ukai s recent paper see y morimoto s ukai gevrey smoothing effect of solutions for spatially homogeneous nonlinear boltzmann equation without angular cutoff j pseudo differ oper appl 1 2010 139 159 but we extend the range of the index satisfying 2 s 1 1 s 0 1 2 and in this case we consider the kinetic factor in the form of v v instead of v as morimoto and ukai did before

gevrey hypoellipticity for linear and non linear fokker planck equations abstract this paper studies the gevrey regularity of weak solutions of a class of linear and semi linear fokker planck equations

numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions this paper presents a boundary integral equation method with the adjoint generalized neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions for each canonical region two integral equations are solved before one can approximate the boundary values of the mapping function cauchy s type integrals are used for computing the mapping function and its inverse for interior points this method also works for regions with piecewise smooth boundaries three examples are given to illustrate the effectiveness of the proposed method

the cauchy problem for the inhomogeneous non cutoff kac equation in critical besov space abstract in this work we investigate the cauchy problem for the spatially inhomogeneous non cutoff kac equation if the initial datum belongs to the spatially critical besov space we can prove the well posedness of weak solution under a perturbation framework furthermore it is shown that the solution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable in comparison with the recent result in 18 the gelfand shilov regularity index is improved to be optimal to the best of our knowledge our work is the first one that exhibits smoothing effect for the kinetic equation in besov spaces

regularization of cauchy abstract problem for a coupled system for nonlinear elliptic equations abstract in this paper we consider a cauchy problem of coupled elliptic equations we show that the problem is ill posed in the sense of hadamard and a regularization is in order under some weak a priori assumptions on the exact solution we propose a new regularization method to stabilize the problem when the source term is a global or local lipschitz function furthermore we also obtain error estimates between the regularized solution and the sought solution

smoothing effects for classical solutions of the full landau equation in this work we consider the smoothness of the solutions to the full landau equation in particular we prove that any classical solutions such as the ones obtained by guo in a close to equilibrium setting become immediately smooth with respect to all variables this shows that the landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation

uncertainty principle and kinetic equations abstract a large number of mathematical studies on the boltzmann equation are based on the grad s angular cutoff assumption however for particle interaction with inverse power law potentials the associated cross sections have a non integrable singularity corresponding to the grazing collisions smoothing properties of solutions are then expected on the other hand the uncertainty principle established by heisenberg in 1927 has been developed so far in various situations and it has been applied to the study of the existence and smoothness of solutions to partial differential equations this paper is the first one to apply this celebrated principle to the study of the singularity in the cross sections for kinetic equations precisely we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations in particular we give some estimates on the regularity of solutions in sobolev spaces w r t all variables for both linearized and nonlinear space inhomogeneous boltzmann equations without angular cutoff as well as the linearized space inhomogeneous landau equation

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

a new general filter regularization method for cauchy problems for elliptic equations with a locally lipschitz nonlinear source up to now studies on the semi linear cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global lipschitz function the current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local lipschitz function in such a situation the methods of solution from the previous studies with a global lipschitz source term are not directly applicable and therefore novel ideas and techniques need to be developed to tackle the local lipschitz nonlinearity this locally lipschitz source arises in many applications of great physical interest governed by for example the sine gordon lane emden allen cahn and liouville equations the inverse problem is severely ill posed in the sense of hadamard by violating the continuous dependence upon the input cauchy data therefore in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method under some priori assumptions on the exact solution we prove and obtain rigorously convergence estimates

analytic smoothness effect of solutions for spatially homogeneous landau equation in this paper we study the smoothness e ect of cauchy problem for the spatially homogeneous landau equation in the hard potential case and themaxwellian molecules case we obtain the analytic smoothing e ect for the solutions under rather weak assumptions on the initial datum

multicritical scaling in the magnetic hard square lattice gas the scaling behaviour of the magnetic hard square lattice gas is examined in the vicinity of the multicritical line located by the recent exact solution a simple scaling form is proposed for the singular part of the free energy and corresponding scaling forms derived for the sublattice density difference the magnetisation and the densities these are all shown to be consistent with the exact results when restricted to the exact solution manifold 1 1 the case of the densities this is achieved by proving appropriate elliptic function identities

on an inverse boundary value problem of a nonlinear elliptic equation in three dimensions abstract this work considers an inverse boundary value problem for a 3d nonlinear elliptic partial differential equation in a bounded domain in general the problem is severely ill posed the formal solution can be written as a hyperbolic cosine function in terms of the 2d elliptic operator via its eigenfunction expansion and it is shown that the solution is stabilized or regularized if the large eigenvalues are cut off in a theoretical framework a truncation approach is developed to approximate the solution of the ill posed problem in a regularization manner under some assumptions on regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type a local lipschitz case of source term for this nonlinear problem is obtained for numerical illustration two examples on the elliptic sine gordon and elliptic allen cahn equations are constructed to demonstrate the feasibility and efficiency of the proposed methods

boundary integral equations with the generalized neumann kernel for laplace s equation in multiply connected regions this paper presents a new boundary integral method for the solution of laplace s equation on both bounded and unbounded multiply connected regions with either the dirichlet boundary condition or the neumann boundary condition the method is based on two uniquely solvable fredholm integral equations of the second kind with the generalized neumann kernel numerical results are presented to illustrate the efficiency of the proposed method

gelfand shilov and gevrey smoothing effect of the cauchy problem for fokker planck equation abstract in this work we study the cauchy problem for fokker planck equation with the external potential we show that the solution of the cauchy problem enjoys gelfand shilov s 1 1 r 3 regularizing property with respect to the velocity variable and gevrey g 3 2 r 3 regularizing property with respect to the position variable

a solvable lattice model and related rogers ramanujan type identities an exactly solvable lattice model is presented which is a natural generalization of baxter s hard hexagon model to three spin states the free energy and spin densities are calculated related rogers ramanujan type identities are also studied

numerical conformal mapping via the szego kernel abstract a new method to compute the riemann mapping function is numerically implemented and tested on examples the method expresses the szego kernel as the solution of a second kind integral equation the equation has its origin in an earlier non numerical work of stein and one of the authors 8 the experimental results show the algorithm to be effective and stable

continuously varying exponents in magnetic hard squares the commuting row to row transfer matrices of magnetic hard squares on the multicritical t manifold are shown to satisfy a special functional equation called an inversion identity for strip widths up to n 32 these equations are solved numerically for the transfer matrix eigenvalues the central charge or conformal anomaly c 1 is obtained from 1 n2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps the magnetic scaling dimension is xm 1 8 and the sublattice density difference scaling dimension is xe 1 9 2 y where y 2 lambda pi and the interaction dependent crossing parameter lambda varies between 0 and 2 pi 3 scaling dimensions for further operators fall in the sequence xn n2xe where n 1 2 3 4 the critical exponents also vary continuously along the multicritical line and are given by alpha 14 9y 16 9y beta m 9 2 y 16 16 9y beta e 1 2 16 9y nu 9 2 y 2 16 9y and delta 15 these exponents are simply related to the exponents of the critical eight vertex and ashkin teller models which also exhibit z2 z2 symmetry and the authors argue that these models lie in the same universality class

parallel slits map of bounded multiply connected regions abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a parallel slit region the method is based on some uniquely solvable boundary integral equations with adjoint classical adjoint generalized and modified neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method

gelfand shilov and gevrey smoothing effect for the spatially inhomogeneous non cutoff kac equation abstract we consider the spatially inhomogeneous non cutoff kac s model of the boltzmann equation we prove that the cauchy problem for the fluctuation around the maxwellian distribution enjoys gelfand shilov regularizing properties with respect to the velocity variable and gevrey regularizing properties with respect to the position variable

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

numerical conformal mapping via a boundary integral equation with the generalized neumann kernel we present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region g onto the five classical canonical slit domains the method is based on a uniquely solvable boundary integral equation with the generalized neumann kernel using the proposed method the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix the method can be also used for calculating the conformal mappings of simply and doubly connected regions the performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries

linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a disk with circular slits the method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method