relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

a nonlinear elliptic equation arising from gauge field theory and cosmology we study radially symmetric solutions of a nonlinear elliptic partial differential equation in r 2 with critical sobolev growth i e the nonlinearity is of exponential type this problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology our results lead to many interesting implications for the physical problems considered for example for the self dual chern simons theory we are able to conclude that the electric charge magnetic flux or energy of a non topological n vortex solution may assume any prescribed value above an explicit lower bound for the einstein matter gauge equations we find a necessary and sufficient condition for the existence of a self dual cosmic string solution such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant

analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

mixed type solutions of the su 3 models on a torus in this paper we study mixed type solutions of su 3 chern simons system see 1 4 below on a two dimensional flat torus nolasco and tarantello commun math phys 213 599 639 2000 among other things nolasco and tarantello obtained solutions of 1 4 as minimizers of several functionals closely related to 1 4 and showed that if n 1 n 2 1 then one of those minimizers turns out to be a mixed type solution that is one component tends to ln frac 1 2 pointwise a e and the other component converges to a solution of a mean field equation we call these kinds of solutions mixed type i solutions in this paper we prove two main results i the asymptotic analysis of mixed type i solutions with arbitrary configuration of vortex points and ii the existence of mixed type i solutions under a non degenerate condition this non degenerate condition also ensures some uniqueness result in particular our results imply that when n 1 n 2 1 there are only two mixed type i solutions of 1 4

on a system of nonlinear elliptic equations arising in theoretical physics in this paper we study a system of nonlinear elliptic equations known as the vortex equations in 2 dimensions arising from the field theoretical descriptions of several models in physics when the underlying space is a closed surface we prove the existence and uniqueness of a solution under a necessary and sufficient condition when the space is r2 we establish the existence uniqueness and sharp decay estimates for a solution

the relativistic non abelian chern simons equations we study ther xr system of nonlinear elliptic equations delta u a lambda sum limits b 1 r k ab e u b lambda sum limits b 1 r sum limits c 1 r e u b k ab e u c k bc 4 pi sum limits j 1 n a delta p aj a 1 2 r x r 2 where 0 is a constant parameter k kab is the cartan matrix of a semi simple lie algebra and p is the dirac measure concentrated atp r 2 this system of equations arises in the relativistic non abelian chern simons theory and may be viewed as a nonintegrable deformation of the integrable toda system we establish the existence of a class of solutions known as topological multivortices the crucial step in our method is the use of the decomposition theorem of cholesky for positive definite matrices so that a variational principle can be formulated

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

bubbling string solutions for the self dual einstein maxwell higgs equation abstract in this article we establish the existence of type i nontopological bubbling solutions for the einstein maxwell higgs equation in r 2 our solutions bubble at some string points and the local profile near those points comes from the radially symmetric solutions we also prove that such bubbling solutions cannot exist on compact surfaces

uniqueness of topological solutions and the structure of solutions for the chern simons system with two higgs particles the existence of topological solutions for the chern simons equation with two higgs particles has been proved by lin ponce and yang 16 however both the uniqueness problem and the existence of non topological solutions have been left open in this paper we consider the case of one vortex at origin among others we prove the uniqueness of topological solutions and give a complete study of the radial solutions in particular the existence of some non topological solutions

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

a system of elliptic equations arising in chern simons field theory we prove the existence of topological vortices in a relativistic self dual abelian chem simons theory with two higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over r 2 we present two approaches to prove existence of solutions on bounded domains via minimization of an indefinite functional and via a fixed point argument we then show that we may pass to the full r 2 limit from the bounded domain solutions to obtain a topological solution in r 2

minimal positive stencils in meshfree finite difference methods for the poisson equation meshfree finite difference methods for the poisson equation approximate the laplace operator on a point cloud desirable are positive stencils i e all neighbor entries are of the same sign classical least squares approaches yield large stencils that are in general not positive we present an approach that yields stencils of minimal size which are positive we provide conditions on the point cloud geometry so that positive stencils always exist the new discretization method is compared to least squares approaches in terms of accuracy and computational performance

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

blow up analysis for boltzmann poisson equation in onsager s theory for point vortices with multi intensities abstract in this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one sided borel probability measure if such a sequence blows up we derive some estimate which is related to the behavior of solution near the blow up point moreover we study the two intensities case to consider the sufficient condition for this estimate our main results are new for the standard mean field equation as well

generalized bc systems based on hermitian vector bundles a generalized bc system associated to a hermitian vector bundle over a riemann surface is introduced in close analogy to the usual rank one case some of the geometric analogies to the well known case are studied in particular if there are no zero modes the non abelian theta divisor appears in the general case where zero modes exist it seems to be more difficult to find a natural description it is discussed in detail why no such system exists in low genus that is on the riemann sphere and on elliptic curves

on the solutions of liouville systems for some prescribed m 0 together with the dirichlet boundary condition u 0 on 0 the liouville equation finds applications in various fields of physics and mathematics to name a few it represents the newtonian potential of a cluster of self gravitating mass distribution where a 0 1 6 36 37 the electric potential induced by charge carriers in the theory of electrolytes where a 0 31 and maximal entropy solutions of the incompressible euler equation 23 9 in the field of differential geometry it stands for the problem of finding a metric corresponding to a prescribed gaussian curvature 3 10 the system analog of l takes the form

on the chern simons higgs vortex equation without gauge fields abstract this paper is concerned with the nonself dual chern simons higgs model on r 2 with vanishing gauge fields we prove the existence of radial solutions with the topological boundary condition and the nonexistence of radial solutions with the nontopological boundary condition we also establish the asymptotic properties of solutions and derive the quantization of the potential energy

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

the existence of chern simons vortices a new type of vortices called chern simons vortices is considered and the existence is established

solutions of the generalized bogomol nyi equations via monotone iterations this paper is concerned with the numerical solutions of the classical abelian higgs vortex model in r2 in the critical coupling phase a globally convergent monotone iterative method will be presented to approximate finite energy multivortex solutions of both the classical and the generalized bogomol nyi equations in the latter context it is illustrated through a series of numerical examples that the higgs potential density function may be adjusted to realize in a wide range fairly different magnetic concentration pictures of the model

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

algebraic theory of multiplicative schwarz methods the convergence of multiplicative schwarz type methods for solving linear systems when the coefficient matrix is either a nonsingular m matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings the effect on convergence of algorithmic parameters such as the number of subdomains the amount of overlap the result of inexact local solves and of coarse grid corrections global coarse solves is analyzed in an algebraic setting results on algebraic additive schwarz are also included

uniqueness and solution structure of nonlinear equations arising from the chern simons gauged o 3 sigma models abstract in this paper we study radially symmetric solutions of nonlinear equations arising from the chern simons gauged o 3 sigma models there are three types of solutions according to the boundary conditions topological solutions nontopological solutions of type i and ii we classify radially symmetric nontopological solutions of type i and ii by the decay rates at infinity in particular we provide a complete picture of the graph of which gives uniqueness and structure of solutions

existence and uniqueness of topological multivortex solutions of the self dual chern simons cp 1 model abstract in this paper we study the existence and uniqueness of the topological chern simons vortices in the cp 1 model in r 2 after reducing the self dual equations to semilinear elliptic partial differential equations we show that a topological solution exists and it is unique up to a real smooth function if the chern simons coupling constant is sufficiently small

asymptotics for the condensate multivortex solutions in the self dual chern simons cp 1 model in this paper we study the asymptotics for the condensate multivortex solutions in the self dual chern simons cp 1 model when the breaking parameter s belongs to 12 12 we show that for any sequence of multivortex solutions which lies between suitable super and subsolutions with respect to the chern simons coupling constant we can find a subsequence which converges to a constant depending only on s as goes to zero also we investigate the locally uniform convergence speed

classification of solutions of elliptic equations arising from a gravitational o 3 gauge field model abstract in this paper we study an elliptic equation arising from the self dual maxwell gauged o 3 sigma model coupled with gravity when the parameter equals 1 and there is only one singular source we consider radially symmetric solutions there appear three important constants a positive parameter a representing a scaled gravitational constant a nonnegative integer n 1 representing the total string number and a nonnegative integer n 2 representing the total anti string number the values of the products a n 1 a n 2 0 play a crucial role in classifying radial solutions by using the decay rates of solutions at infinity we provide a complete classification of solutions for all possible values of a n 1 and a n 2 this improves previously known results

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

an algebraic convergence theory for restricted additive schwarz methods using weighted max norms convergence results for the restrictive additive schwarz ras method of cai and sarkis siam j sci comput 21 1999 pp 792 797 for the solution of linear systems of the form ax b are provided using an algebraic view of additive schwarz methods and the theory of multisplittings the linear systems studied are usually discretizations of partial differential equations in two or three dimensions it is shown that in the case of a symmetric positive definite the projections defined by the methods are not orthogonal with respect to the inner product defined by a and therefore the standard analysis cannot be used here the convergence results presented are for the class of m matrices and more generally for h matrices using weighted max norms comparison between different versions of the ras method are given in terms of these norms a comparison theorem with respect to the classical additive schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory several ras variants are considered including new ones and two level schemes

overlapping blocks by growing a partition with applications to preconditioning starting from a partitioning of an edge weighted graph into subgraphs we develop a method which enlarges the respective sets of vertices to produce a decomposition with overlapping subgraphs the vertices to be added when growing a subset are chosen according to a criterion which measures the strength of connectivity with this subset by using our method on the directed graph associated with a matrix we obtain an overlapping decomposition of the set of variables which can be used for algebraic additive and multiplicative schwarz preconditioners we present a complexity analysis of this block growing method thus proving its computational efficiency numerical results for problems stemming from various application areas show that with these overlapping schwarz preconditioners we usually substantially improve gmres convergence as compared to preconditioners based on a nonoverlapping decomposition or an overlapping decomposition based on level sets without other criteria as well as incomplete lu

existence theorems for a general 2 2 non abelian chern simons higgs system over a torus in this paper we study a general 2 2 non abelian chern simons higgs system of the form ui 1e2 j 12kjieuj j 12 k 12kkjkjieujeuk 4 j 1ni pij x i 1 2 over a flat 2 torus t2 where e 0 p is the dirac measure at p ni n i 1 2 k is a non degenerate 2 2 matrix of the form k 1 a a b1 b which may cover the physically interesting case when k is a cartan matrix of a rank 2 semisimple lie algebra concerning the existence results of this type system over t2 usually in the literature there is a requirement that a b 0 however it is an open problem so far for the solvability about such system with a b 0 such that this system admits a solution over the torus if 0 e e0 provided a b are suitably small furthermore if ab 0 in addition with suitable condition on a b n1 n2 this system admits a mountain pass solution our argument is based on a perturbation approach and the mountain pass lemma

uniqueness of topological multivortex solutions in the maxwell chern simons model abstract in this paper we prove the uniqueness of topological multivortex solutions for the self dual maxwell chern simons u 1 model if the chern simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large on the other hand we also establish the sharp region of the flux for non topological solutions and provide the classification of radial solutions of all types in the case of one vortex point

new blow up phenomena for su n 1 toda system abstract we consider the su n 1 toda system s u 1 2 e u 1 e u 2 e u k 0 in u 2 e u 1 2 e u 2 e u k 0 in u k e u 1 e u 2 2 e u k 0 in u 1 u 2 u k 0 on if 0 and is symmetric with respect to the origin we construct a family of solutions u 1 u k to s such that the i th component u i blows up at the origin with a mass 2 i 1 as goes to zero

nontopological n vortex condensates for the self dual chern simons theory we prove the existence of nontopological n vortex solutions for an arbitrary number n of vortex points for the self dual chern simons higgs theory with t hooft periodic boundary conditions we use a shadowing type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the liouville equation 2003 wiley periodicals inc

existence of the self graviting chern simons vortices we prove existence of multivortex solutions of the self dual einstein chern simons higgs system proposed by clement phys rev d 54 1844 1847 1996 we consider both the topological and the nontopological boundary conditions for open conformally flat manifolds for nontopological boundary conditions we use perturbation argument from a solution of the liouville equation combined with the implicit function theorem using this argument we have existence for arbitrary positive number for the gravitational constant for topological boundary condition we construct solutions for small gravitational constant by using the super subsolution method for sufficiently large gravitational constant we have a nonexistence result for the radially symmetric topological solutions we also obtain the decay estimates near infinity for both of the topological and the nontopological solutions

evaluating solutions on an elliptic problem in a gravitational gauge field theory abstract an elliptic equation arising from the study of static solutions with prescribed zeros and poles of the einstein equations coupled with the classical sigma model and an abelian gauge field is considered we classify the solutions and establish the uniqueness of radially symmetric solutions we also complete a classification of symmetric solutions of an elliptic equation on the sphere

existence of multivortex solutions in the self dual chern simons higgs theory in a background metric in this paper we show the existence of the multiple multivortex solutions of the self duality equations of 2 1 dimensional chern simons higgs model in a background metric of the form g diag 1 b x b x with b x o x l as x for some l 2

uniqueness and symmetry results for solutions of a mean field equation on 2 via a new bubbling phenomenon motivated by the study of gauge field vortices we consider a mean field equation on the standard two sphere involving a dirac distribution supported at a point p of s 2 as needed for the applications we show that solutions concentrate exactly at p for some limiting value of a given parameter we use this fact to obtain symmetry about the axis op and uniqueness property for the solution the presence of the dirac measure makes such a task particularly delicate indeed we need to rule out the possibility that after blow up in a suitable scale the solution sequence may admit a double peak profile described in terms of appropriate limiting problems for instance we need to account for the presence of non axially symmetric solutions in such limiting problems in this process we establish a symmetry result about a maximal circle through p and its antipodal point p that applies to more general situations where the full axial symmetry cannot be expected

existence and properties of radial solutions in the self dual chern simons o 3 sigma model in this paper we study the self dual equations arising from the chern simons gauged o 3 sigma model with symmetric potential we prove the existence of radially symmetric solutions of the reduced elliptic equation having topological and nontopological boundary conditions

uniqueness and structure of solutions to the dirichlet problem for an elliptic system abstract in this paper we consider the dirichlet problem for an elliptic system on a ball in r 2 by investigating the properties for the corresponding linearized equations of solutions and adopting the pohozaev identity and implicit function theorem we show the uniqueness and the structure of solutions

periodic maxwell chern simons vortices with concentrating property in order to study electrically and magnetically charged vortices in fractional quantum hall effect and anyonic superconductivity the maxwell chern simons mcs model was introduced by lee et al phys lett b 252 79 83 1990 as a unified system of the classical abelian higgs model ah and the chern simons cs model in this article the first goal is to obtain the uniform cs limit result of mcs model with respect to the chern simons parameter without any restriction on either a particular class of solutions or the number of vortex points as the chern simons mass scale tends to infinity the most important step for this purpose is to derive the relation between the higgs field and the neutral scalar field our cs limit result also provides the critical clue to answer the open problems raised by ricciardi and tarantello comm pure appl math 53 811 851 2000 and tarantello milan j math 72 29 80 2004 and we succeed to establish the existence of periodic maxwell chern simons vortices satisfying the concentrating property of the density of superconductive electron pairs furthermore we expect that the cs limit analysis in this paper would help to study the stability multiplicity and bubbling phenomena for solutions of the mcs model

blow up analysis for a cosmic strings equation abstract in this paper we develop a blow up analysis for solutions of an elliptic pde of liouville type over the plane such solutions describe the behavior of cosmic strings parallel in a given direction for a w boson model coupled with einstein s equation we show how the blow up behavior of the solutions is characterized according to the physical parameters involved by new and surprising phenomena for example in some cases after a suitable scaling the blow up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the uniformization of the riemann sphere with conical singularities

a necessary and sufficient condition for the existence of multisolitons in a self dual gauged sigma model this paper presents a resolution of the gaugedo 3 sigma model proposed by b j schroers in which the matter field o mapsr2 intos2 while the vector gauge potential gives rise to a magnetic field it is shown that for each natural numbern there are solutions to saturate the classical energy lower bounde 4 n for the field configurations in the topological family deg o n if and only ifn 1 furthermore the solutions obtained depend on at least 4n 3 continuous parameters the associated magnetic flux can assume its value in an open interval and the decay rates of the field strengths may be specified in a suitable range these solutions are multisolitons represented byn prescribed lumps of the magnetic field simulatingn identical particles in equilibrium and are governed by a nonlinear elliptic equation with both vortex and anti vortex source terms

non topological multivortex solutions to the self dual maxwell chern simons higgs systems in this paper we construct non topological multivortex solutions to the non relativistic self dual maxwell chern simons higgs system in r2 which make the energy functional finite moreover our proof of the existence of solutions reveals precise asymptotic behavior of solutions near spatial infinity using exactly the same method we also establish the existence of non topological multivortex solutions to the relativistic self dual maxwell chern simons higgs system

multiplicity results for the assigned gauss curvature problem in r2 to study the problem of the assigned gauss curvature with conical singularities on riemanian manifolds we consider the liouville equation with a single dirac measure on the two dimensional sphere by a stereographic projection we reduce the problem to a liouville equation on the euclidean plane we prove new multiplicity results for bounded radial solutions which improve on earlier results of c s lin and his collaborators based on numerical computations we also present various conjectures on the number of unbounded solutions using symmetries some multiplicity results for non radial solutions are also stated

statistical mechanics approach to some problems in conformal geometry a weak law of large numbers is established for a sequence of systems of n classical point particles with logarithmic pair potential in rn or sn n n which are distributed according to the configurational microcanonical measure e h or rather some regularization thereof where h is the configurational hamiltonian and e the configurational energy when n with non extensive energy scaling e n2e the particle positions become i i d according to a self consistent boltzmann distribution respectively a superposition of such distributions the self consistency condition in n dimensions is some nonlinear elliptic pde of order n pseudo pde if n is odd with an exponential nonlinearity when n 2 this pde is known in statistical mechanics as poisson boltzmann equation with applications to point vortices 2d coulomb and magnetized plasmas and gravitational systems it is then also known in conformal differential geometry where it is the central equation in nirenberg s problem of prescribed gaussian curvature for constant gauss curvature it becomes liouville s equation which also appears in two dimensional so called quantum liouville gravity the pde for n 4 is paneitz equation and while it is not known in statistical mechanics it originated from a study of the conformal invariance of maxwell s electromagnetism and has made its appearance in some recent model of four dimensional quantum gravity in differential geometry the paneitz equation and its higher order n generalizations have applications in the conformal geometry of n manifolds but no physical applications yet for general n interestingly though all the paneitz equations have an interpretation in terms of statistical mechanics

existence of mixed type solutions in the chern simons gauge theory of rank two in r2 abstract we consider the self dual equations arising from the chern simons gauge theory of rank 2 such as the s u 3 s o 5 and g 2 chern simons model in r 2 there are three possible types of solutions in these theories namely topological nontopological and mixed type solutions the existence of a mixed type solution for an arbitrary configuration of vortex points has been a long standing open problem we construct here a family of mixed type solutions u 1 e u 2 e e 0 for an arbitrary configuration of vortex points under a mild non degeneracy condition the main new idea of our construction of an approximate solution is to use different scalings for different components in the process of the finite dimensional reduction all estimates for this particular construction become rather delicate

existence of topological multi string solutions in abelian gauge field theories in this paper we consider a general form of self dual equations arising from abelian gauge field theories coupled with the einstein equations by applying the super subsolution method we prove that topological multi string solutions exist for any coupling constant which improves previously known results we provide two examples for application the self dual einstein maxwell higgs model and the gravitational maxwell gauged o 3 sigma model

uniqueness of the topological multivortex solution in the self dual chern simons theory we establish a uniqueness result for the topological multivortex solution to the self dual equations of the abelian relativistic self dual chern simons higgs model we prove that the topological multivortex solution is unique if the chern simons coupling parameter 0 is sufficiently small we also establish a uniqueness result for 0 sufficiently large

nontopological solutions in the self dual maxwell chern simons gauged o 3 sigma model abstract in this paper we prove the existence of nontopological solutions in the self dual maxwell chern simons gauged o 3 sigma model in the plane if the scaling parameter is small enough the reduced equation can be regarded as a perturbation of the liouville equation then we use the generalized implicit function theorem to prove the existence of solutions with nontopological boundary conditions using this result we show the existence of symmetrically topological solutions and nontopological solutions for the original self dual equations whose magnetic fluxes are slightly bigger than the critical value moreover we also compute the topological charge and the degree of the spin vectors

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

uniqueness of topological solutions for a class of self dual vortex theories we study the solutions of topological type for a class of self dual vortex theories in two dimensions we consider the regime corresponding to the limit of small vortex core size with respect to the separation distance between vortices namely as the scaling parameter 0 tends to zero using a gluing technique for the corresponding nonlinear elliptic equation on the plane with any number finite or countable of prescribed singular sources we prove the existence of multi vortex solutions which behave as a single vortex solution near each vortex point up to an error exponentially small as 0 moreover in the physically relevant cases namely when the vortex points are either finite or periodically arranged in the plane we prove that the multi vortex solution satisfying a topological condition is unique for 0 sufficiently small

classical solutions in the born infeld theory locally concentrated solutions in the born infeld theory are presented in particular existence and uniqueness theorems are established for multicentred magnetic string solutions induced from a higgs field over a closed riemann surface or a euclidean plane on any given compact surface the born infeld parameter may be adjusted under a necessary and sufficient condition to allow the existence of an arbitrarily large number of strings

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

degree counting for toda system with simple singularity one point blow up abstract in this paper we study the degree counting formula of the rank two toda system with simple singular source when 1 0 4 4 8 and 2 4 n the key step is to derive the degree formula of the shadow system which arises from the bubbling solutions as 1 tends to 4 in order to compute the topological degree of the shadow system we need to find some suitable deformation during this deformation we shall deal with new difficulty arising from the phenomenon blow up does not necessarily imply concentration of mass this phenomenon occurs due to the collapsing of singularities this is a continuation of the previous work 25

existence and uniqueness of generalized vortices we investigate properties of the static noninteracting vortices determined by equations which generalize the first order ginzburg landau equations we prove that for each set of n points in the plane a unique solution exists to the first order equations with vortex number n these n points mark the positions of the n vortices and are the only points at which the higgs field vanishes regularity properties of the solution are related to those of an arbitrary non negative function in the theory

hecke transformation and the generalized theta function we describe the behavior of the generalized theta function with respect to the hecke transformations let e be a holomorphic vector bundle over rank r and degree r g 1 over a compact riemann surface x of genus g let p e respectively p e be the space of all lines respectively hyperplanes in the fibers of e certain zariski open subsets of p e p e and p e p e p e p e parametrize holomorphic families of vector bundles over x of rank r and degree r g 1 we describe the generalized theta line bundle for these families

a sup inf inequality near r 0 let be a domain in r2 not necessarily bounded consider the semi linear elliptic equation u r x eu x we prove that for any compact subset k of there is a constant c such that the inequality 1 supku inf u c holds for all solutions u this type of inequality was first established by brezis li and shafrir h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 under the assumption that r x is positive and bounded away from zero it has become a useful tool in estimating the solutions of semi linear elliptic equations either in euclidean spaces or on riemannian manifolds see h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 c c chen c s lin a sharp sup inf inequality for a nonlinear elliptic equation in r2 comm anal geom 6 1998 1 19 w chen c li gaussian curvature in the negative case proc amer math soc 131 2003 741 744 w chen c li indefinite elliptic problems with critical exponent in advances in non linear pde and related areas world scientific 1998 pp 67 79 y y li i shafrir blow up analysis for solutions of u veu in dimension two indiana univ math j 43 1994 1255 1270 in brezis li and shafrir s result the constant c depends on the lower bound of the function r x in this paper we remove this restriction and extend the inequality to the case where r x is allowed to have zeros so that it can be applied to obtain a priori estimates for a broader class of equations as we will illustrate in the last section the key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in r2 which is interesting and useful in its own right

the conformal plate buckling equation the linear equation 2 u 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over r 2 has the particularly interesting nonlinear generalization g 2 u 1 where ag e 2u is the laplace beltrami operator for the metric g e 2u g 0 with g 0 the standard euclidean metric on r 2 this conformal elliptic pde of fourth order is equivalent to the nonlinear system of elliptic pdes of second order u x k g x exp 2u x 0 and k g x exp 2u x 0 with x r 2 describing a conformally flat surface with a gauss curvature function k g that is generated self consistently through the metric s conformal factor we study this conformal plate buckling equation under the hypotheses of finite integral curvature f kg exp 2u dx k finite area exp 2u dx a and the mild compactness condition k l 1 b 1 y uniformly with respect to y r 2 we show that asymptotically for x all solutions behave like u x 2 ln x c o 1 and k x 2 ln x c o 1 with 2 4 and 2 4 we also show that for each k 2 4 there exists a k and a radially symmetric solution pair u k satisfying k u k and max k k which is unique modulo translation of the origin and scaling of x coupled with a translation of u

on patlak keller segel system for several populations a gradient flow approach abstract we study the global in time existence of solutions to the parabolic elliptic patlak keller segel system of multi species populations we prove that if the initial mass satisfies an appropriate notion of sub criticality then the system has a solution defined for all time we explore the gradient flow structure in the wasserstein space to study the question of existence moreover we show that the obtained solution satisfies energy dissipation inequality

on a planar liouville type problem in the study of selfgravitating strings abstract motivated by the construction of selfgravitating strings cf yang 2001 1994 22 23 we analyze a liouville type equation on the plane derived in yang 1994 23 we establish sharp existence and uniqueness properties for the corresponding radial solutions we investigate also when the problem allows for non radial solutions

multivortex solutions in the chern simons gauged o 3 sigma model on a doubly periodic domain abstract we establish the existence and uniqueness results for multivortex solutions of an elliptic equation arising from the self dual chern simons gauged o 3 sigma model with a symmetric potential on a flat 2 torus we prove that if the parameter e 0 is small then the elliptic governing equation admits a maximal solution we also study the asymptotic behavior of the maximal solutions as e 0 by using this asymptotic behavior we establish the uniqueness results for solutions which tend to a e as e 0 under suitable conditions on the dirac measure

extensions of certain graph based algorithms for preconditioning the original tpablo algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries this block diagonal can then be used as a preconditioner we propose and analyze three extensions of this approach we incorporate a nonsymmetric permutation to obtain a large diagonal we use a more general parametrization for tpablo and we use a block gauss seidel preconditioner which can be implemented to have the same execution time as the corresponding block jacobi preconditioner experiments are presented showing that for certain classes of matrices the block gauss seidel preconditioner used with the system permuted with the new algorithm can outperform the best ilut preconditioners in a large set of experiments

liouville theorems for kirchhoff equations in rn this paper is devoted to the nonexistence of nontrivial weak solutions for the kirchhoff equation a b rn u 2 dx u x qf u in rn we prove that the equation has no weak solution if a 0 b 0 q 2 and f is a positive convex nondecreasing function if only b 0 and f is a non negative function we establish the nonexistence of weak solutions u satisfying lim infr r1 n2 br x qf u dx 0 this implies that the equation has no weak solution when n 2 and f is a positive function we also show that the equation has no stable weak solution in dimension n 0

existence and asymptotics of topological solutions in the self dual maxwell chern simons o 3 sigma model we are concerned with the topological vortex equations arising in the self dual maxwell chern simons o 3 sigma model we prove the existence of solutions for any parameters and q by a variational method we also verify both the maxwell limit and the chern simons limit for the variational solutions