sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

on liouville systems at critical parameters part 1 one bubble abstract in this paper we consider bubbling solutions to the general liouville system 0 1 g u i k j 1 n a i j j k h j e u j k h j e u j k 1 0 in m i 1 n n 2 where m g is a riemann surface and a a i j n n is a constant non negative matrix and j k j as k among other things we prove the following sharp estimates 1 the location of the blowup point 2 the convergence rate of j k j j 1 n these results are of fundamental importance for constructing bubbling solutions it is interesting to compare the difference between the general liouville system and the su 3 toda system on estimates 1 and 2

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

existence of bubbling solutions for chern simons model on a torus we study the existence of bubbling solutions for the the following chern simons higgs equation delta u frac1 varepsilon 2 rm e u 1 rm e u 4 pi sum i 1 2k delta p i quad text in omega where is a torus if k 1 for any critical point q of the associated sum of the green functions we introduce a quantity d q see 1 11 below we show that for any non degenerate critical point q with d q 0 the above problem has a solution ue satisfying that e 0 ue blows up at q the calculations in this paper also show that if a sequence of solutions ue blows up at q as e 0 then q must be a critical point of the associated sum of the green functions and d q leqq 0 so the condition d q 0 is almost necessary to obtain our result we also construct solutions with k bubbles for k geqq 2

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

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

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

on the mean field type bubbling solutions for chern simons higgs equation abstract this paper is the second part of our comprehensive study on the structure of the solutions for the following chern simons higgs equation 0 1 u 1 e 2 e u 1 e u 4 j 1 n p j in u is doubly periodic on where is a parallelogram in r 2 and e 0 is a small parameter in part 1 29 we proved the non coexistence of different bubbles in the bubbling solutions and obtained an existence result for the chern simons type bubbling solutions under some nearly necessary conditions mean field type bubbling solutions for 0 1 have been constructed in 27 in this paper we shall study two other important issues for the mean field type bubbling solutions the necessary conditions for the existence and the local uniqueness the results in this paper lay the foundation to find the exact number of solutions for 0 1

singular mean field equations on compact riemann surfaces for a general class of elliptic pde s in mean field form on compact riemann surfaces with exponential nonlinearity we address the question of the existence of solutions with concentrated nonlinear term which in view of the applications are physically of definite interest in the model we also include the possible presence of singular sources in the form of dirac masses which makes the problem more difficult to attack

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

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

elliptic functions green functions and the mean field equations on tori we show that the green functions on flat tori can have either 3 or 5 critical points only there does not seem to be any direct method to attack this problem instead we have to employ sophisticated non linear partial differential equations to study it we also study the distribution of number of critical points over the moduli space of flat tori through deformations the functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori the general picture is also emerged though some of the necessary technicality is still to de developed 1 introduction and statement of results the study of geometric or analytic problems on two dimensional tori is the same as the study of problems on r 2 with doubly periodic data such situations occur naturally in sciences and mathematics since early days the mathematical foundation of elliptic functions was subsequently devel oped in the 19th century it turns out that these special functions are rather deep objects by themselves tori of different shape may result in very dif ferent behavior of the elliptic functions and their associated objects arith metic on elliptic curves is perhaps the eldest and the most vivid example in this paper we show that this is also the case for certain non linear par tial differential equations indeed researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity this leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case however this turns out to be false we will show that the solvability of themean field equation dependson the shape of the green function which in turn depends on the geometry of the tori in an essential way recall that the green function g z w on a flat torus t c z 1 z 2 is the unique function on t t which satisfies zg z w w z 1 t

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

a topological degree counting for some liouville systems of mean field type let a a ij n n be an invertible matrix and a 1 a ij n n be the inverse of a in this paper we consider the generalized liouville system n 0 1 g u i n a ij j h j e u j h j e u j 1 0 in m j 1 where 0 h j c 1 m and j and prove that under the assumptions of h 1 and h 2 see introduction the leray schauder degree of 0 1 is equal to m 1 m n n if 1 n satisfies 8 n n n i 1 i 1 i j n a ij i j 8 n 1 n n i 1 i equation 0 1 is a natural generalization of the classic liouville equation and is the euler lagrangian equation of the nonlinear function u 1 2 m 1 i j n a ij g u i g u j n n i 1 m i u i n n i 1 i log m h i e u i the liouville system 0 1 has arisen in many different research areas in mathematics and physics our counting formulas are the first result in degree theory for liouville systems

limit theorems for the laziest minimal random walk model of elephant type we consider a minimal model of one dimensional discrete time random walk with step reinforcement introduced by harbola kumar and lindenberg 2014 the walker can move forward never backward or remain at rest for each n 1 2 ldots a random time u n between 1 and n is chosen uniformly and if the walker moved forward resp remained at rest at time u n then at time n 1 it can move forward with probability p resp q or with probability 1 p resp 1 q it remains at its present position for the case q 0 several limit theorems are obtained by coletti gava and de lima 2019 in this paper we prove limit theorems for the case q 0 where the walker can exhibit all three forms of asymptotic behavior as p is varied as a byproduct we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees

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

first encounters on combs we consider two random walkers embedded in a finite two dimension comb and we study the mean first encounter time mfet evidencing mainly numerically different scalings with the linear size of the underlying network according to the initial position of the walkers if one of the two players is not allowed to move then the first encounter problem can be recast into a first passage problem mfpt for which we also obtain exact results for different initial configurations by comparing mfet and mfpt we are able to figure out possible search strategies and in particular we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting while for a random setting on average letting one player rest would slow down the search

elephant random walks and their connection to polya type urns in this paper we explain the connection between the elephant random walk erw and an urn model a la polya and derive functional limit theorems for the former the erw model was introduced in phys rev e 70 045101 2004 10 1103 physreve 70 045101 to study memory effects in a highly non markovian setting more specifically the erw is a one dimensional discrete time random walk with a complete memory of its past the influence of the memory is measured in terms of a memory parameter p between zero and one in the past years a considerable effort has been undertaken to understand the large scale behavior of the erw depending on the choice of p here we use known results on urns to explicitly solve the erw in all memory regimes the method works as well for erws in higher dimensions and is widely applicable to related models

green function painleve vi equation and eisenstein series of weight one we study the problem how many singular points of a solution t to the painleve vi equation with parameter 8 1 8 1 8 3 8 might have in c 0 1 here t0 c 0 1 is called a singular point of t if t0 0 1 t0 based on hitchin s formula we explore the connection of this problem with green function and the eisenstein series of weight one among other things we prove i there are only three solutions which have no singular points in c 0 1 ii for a special type of solutions called real solutions here any branch of a solution has at most two singular points in particular at most one pole in c 0 1 iii any riccati solution has singular points in c 0 1 iv for each n 5 and n 6 6 we calculate the number of the real j values of zeros of the eisenstein series en 1 k1 k2 of weight one where k1 k2 runs over 0 n 1 2 with gcd k1 k2 n 1 the geometry of the critical points of the green function on a flat torus e as varies in the modulim1 plays a fundamental role in our analysis of the painleve iv equation in particular the conjectures raised in 22 on the shape of the domain 5 m1 which consists of tori whose green function has extra pair of critical points are completely solved here

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

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

random recursive trees and the elephant random walk one class of random walks with infinite memory so called elephant random walks are simple models describing anomalous diffusion we present a surprising connection between these models and bond percolation on random recursive trees we use a coupling between the two models to translate results from elephant random walks to the percolation process we calculate besides other quantities exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation we further introduce another model the skew elephant random walk and calculate the first and second moment of this process

memory induced anomalous dynamics emergence of diffusion subdiffusion and superdiffusion from a single random walk model we present a random walk model that exhibits asymptotic subdiffusive diffusive or superdiffusive behavior in different parameter regimes this appears to be an instance of a single random walk model leading to all three forms of behavior by simply changing parameter values furthermore the model offers the great advantage of analytical tractability our model is non markovian in that the next jump of the walker is probabilistically determined by the history of past jumps it also has elements of intermittency in that one possibility at each step is that the walker does not move at all this rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior

convergence rate location and z2 condition for fully bubbling solutions to su n 1 toda systems abstract it is well known that the study of su n 1 toda systems is important not only to chern simons models in physics but also to the understanding of holomorphic curves harmonic sequences or harmonic maps from riemann surfaces to c p n one major goal in the study of su n 1 toda system on riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions in this article we use a unified approach to study fully bubbling solutions to general su n 1 toda systems and we prove three major sharp estimates important for constructing bubbling solutions the closeness of blowup solutions to entire solutions the location of blowup points and a z 2 condition

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

uniqueness of bubbling solutions with collapsing singularities abstract the seminal work 7 by brezis and merle showed that the bubbling solutions of the mean field equation have the property of mass concentration recently lin and tarantello in 30 found that the bubbling implies mass concentration phenomena might not hold if there is a collapse of singularities furthermore a sharp estimate 23 for the bubbling solutions has been obtained in this paper we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs the main difficulty comes from that after re scaling the difference of two solutions locally converges to an element in the kernel space of the linearized operator it is well known that the kernel space is three dimensional so the main technical ingredient of the proof is to show that the limit after re scaling is orthogonal to the kernel space

on the multi dimensional elephant random walk the purpose of this paper is to investigate the asymptotic behavior of the multi dimensional elephant random walk merw it is a non markovian random walk which has a complete memory of its entire history a wide range of literature is available on the one dimensional erw surprisingly no references are available on the merw the goal of this paper is to fill the gap by extending the results on the one dimensional erw to the merw in the diffusive and critical regimes we establish the almost sure convergence the law of iterated logarithm and the quadratic strong law for the merw the asymptotic normality of the merw properly normalized is also provided in the superdiffusive regime we prove the almost sure convergence as well as the mean square convergence of the merw all our analysis relies on asymptotic results for multi dimensional martingales

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

a martingale approach for the elephant random walk the purpose of this paper is to establish via a martingale approach some refinements on the asymptotic behavior of the one dimensional elephant random walk erw the asymptotic behavior of the erw mainly depends on a memory parameter p which lies between zero and one this behavior is totally different in the diffusive regime the critical regime and the superdiffusive regime in the diffusive and critical regimes we establish some new results on the almost sure asymptotic behavior of the erw such as the quadratic strong law and the law of the iterated logarithm in the superdiffusive regime we provide the first rigorous mathematical proof that the limiting distribution of the erw is not gaussian

a priori estimates for d4 and f4 toda systems abstract in this article we establish the a priori bounds of the toda systems with arbitrarily many singular sources associated to the simple lie algebras of type d 4 and f 4 extending the previous works of lin et al 27 20 24 for the a n b n c n and g 2 type lie algebras the problem of obtaining a priori estimates can be reduced to locating the local mass of blowup solutions the key step is to calculate the local mass near each blowup points efficiently a precise estimate on the total mass of entire solutions of toda systems with many singular sources is an indispensable step in our argument

blow up analysis for su 3 toda system abstract using the method of symmetrization and the rescaling we study non compact solution sequence to the su 3 toda system in non abelian relativistic self dual gauge theory i e the quantization of the total mass and classification of the singular limit

pascal principle for diffusion controlled trapping reactions in this paper we analyze the long time behavior of the survival probability p a t of an a particle that performs lattice random walk in the presence of randomly moving traps b we show that for both perfect and imperfect trapping reactions for arbitrary spatial dimension d and for a rather general class of random walks p a t is less than or equal to the survival probability of an immobile target a in the presence of randomly moving traps

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

optimal search strategies of space time coupled random walkers with finite lifetimes we present a simple paradigm for detection of an immobile target by a space time coupled random walker with a finite lifetime the motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed we call these walkers mortal creepers a mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate m while still alive the creeper has a finite mean frequency of change of the direction of motion in particular we consider the efficiency of the target search process characterized by the probability that the creeper will eventually detect the target analytic results confirmed by numerical results show that there is an m dependent optimal frequency opt that maximizes the probability of eventual target detection we work primarily in one dimensional d 1 domains and examine the role of initial conditions and of finite domain sizes numerical results in d 2 domains confirm the existence of an optimal frequency of change of direction thereby suggesting that the observed effects are robust to changes in dimensionality in the d 1 case explicit expressions for the probability of target detection in the long time limit are given in the case of an infinite domain we compute the detection probability for arbitrary times and study its early and late time behavior we further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time we also consider a version of the standard target problem in which many creepers start at random locations at the same time

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

lattice theory of trapping reactions with mobile species we present a stochastic lattice theory describing the kinetic behavior of trapping reactions a stackrel ensuremath rightarrow b b in which both the a and b particles perform an independent stochastic motion on a regular hypercubic lattice upon an encounter of an a particle with any of the b particles a is annihilated with a finite probability finite reaction rate is taken into account by introducing a set of two state random variables char22 gates imposed on each b particle such that an open closed gate corresponds to a reactive passive state we evaluate here a formal expression describing the time evolution of the a particle survival probability which generalizes our previous results we prove that for quite a general class of random motion of the species involved in the reaction process for infinite or finite number of traps and for any time t the a particle survival probability is always larger in the case when a stays immobile than in situations when it moves

sharp existence results for mean field equations with singular data abstract let be a simply connected open and bounded domain in r 2 we are concerned with the nonlinear elliptic problem 0 1 v 8 e v e v 4 j 1 m j p j in v 0 on where j 0 p j denotes the dirac mass with singular point p j and p 1 p m we provide necessary and sufficient conditions for the existence of solutions to 0 1 our result is the two dimensional version of the sharp existence nonexistence result obtained in druet 2002 13 for elliptic equations with critical exponent in dimension 3 in particular we prove that the set m is open where for a given 1 m 0 0 m p 1 p m problem 0 1 has a solution

bubbling solutions for relativistic abelian chern simons model on a torus we prove the existence of bubbling solutions for the following chern simons higgs equation delta u frac 1 varepsilon 2 e u 1 e u 4 pi sum j 1 n delta p j quad rm in omega where is a torus we show that if n 4 and p 1 p j j 2 n then for small e 0 the above problem has a solution u e and as e 0 u e blows up at the vertex point p 1 and satisfies frac 1 varepsilon 2 e u 1 e u to 4 pi n delta p 1 this is the first result for the existence of a solution which blows up at a vertex point

maximum of n independent brownian walkers till the first exit from the half space we consider the one dimensional target search process that involves an immobile target located at the origin and n searchers performing independent brownian motions starting at the initial positions all on the positive half space the process stops when the target is first found by one of the searchers we compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power law tail for large m thus all moments of m up to the order n 1 are finite while the higher moments diverge the prefactor bn increases with n faster than exponentially our solution gives the exit probability of a set of n particles from a box 0 l through the left boundary incidentally it also provides an exact solution of laplace s equation in an n dimensional hypercube with some prescribed boundary conditions the analytical results are in excellent agreement with monte carlo simulations

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

the shadow principle an optimal survival strategy for a prey chased by random predators abstract we consider a lattice model of the annihilation process a b b when a mobile prey a is chased by identical independent predators b performing random motions until one of them finds a and destroys it it is assumed that each predator follows some most probable trajectory around which it performs a random motion it is shown that if the random motion of the predators satisfies certain conditions the prey a can maximize its survival probability by following a specific trajectory which mimics the preferred trajectories of the predators we call this optimal trajectory as the shadow of the predator this is an extension of the so called pascal principle studied in the recent literature we discuss the conditions which allow for such extensions and give examples where they are realized

inverted regions induced by geometric constraints on a classical encounter controlled binary reaction abstract the efficiency of an encounter controlled reaction between two independently mobile reactants on a lattice is characterized by the mean number n of steps to reaction the two reactants are distinguished by their mass with the light walker performing a jump to a nearest neighbor site in each time step while the heavy walker hops only with a probability p we associate p with the temperature of the system to account for geometric exclusion effects in the reactive event two reaction channels are specified for the walkers irreversible reaction occurs either in a nearest neighbor collision or when the two reactants attempt to occupy the same site lattices subject to periodic and to confining boundary conditions are considered for periodic lattices depending on the initial state the reaction time either falls off monotonically with p or displays a local minimum with respect to p occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature such behavior can also occur when one averages over all initial conditions but it disappears if the jump probability of the light walker falls below a characteristic threshold value even more robust behavior can occur on lattices subject to confining boundary conditions depending on the initial conditions the reaction time as a function of p may increase monotonically decrease monotonically display a single maximum or even a maximum and minimum in the latter case one can identify distinct regimes where the above noted inversion in reaction efficiency can occur we document both numerically and theoretically that these inversion regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels and the system s dimensionality and spatial extent our results highlight situations where the description of an encounter controlled reactive event cannot be described by a single effective diffusion coefficient we also distinguish between the inversion region identified here and the marcus inverted region which arises in electron transfer reactions

central limit theorem and related results for the elephant random walk we study the so called elephant random walk erw which is a non markovian discrete time random walk on with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior we prove a law of large numbers and a central limit theorem remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive inside the superdiffusive regime the erw converges to a non degenerate random variable which is not normal we also obtain explicit expressions for the correlations of increments of the erw

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

bubbling solutions for the su 3 chern simons model on a torus we consider the following nonlinear system derived from the su 3 chern simons models on a torus where denotes the dirac measure at when if we look for a solution with then 0 1 is reduced to the chern simons higgs equation the existence of bubbling solutions to 0 1 has been a longstanding problem in this paper we prove the existence of such solutions such that even if 2012 wiley periodicals inc

kinetics of active surface mediated diffusion in spherically symmetric domains we present an exact calculation of the mean first passage time to a target on the surface of a 2d or 3d spherical domain for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion we generalize the results of benichou et al in j stat phys 142 657 2011 and consider a biased diffusion in a general annulus with an arbitrary number of regularly spaced targets on a partially reflecting surface the presented approach is based on an integral equation which can be solved analytically numerically validated approximation schemes which provide more tractable expressions of the mean first passage time are also proposed in the framework of this minimal model of surface mediated reactions we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface

locating the peaks of solutions via the maximum principle ii a local version of the method of moving planes let be a bounded smooth domain in r 2n n 2 the well known moser trudinger inequality ensures the nonlinear functional jp u is bounded from below if and only if 2n 2 2n n n 1 w 2n where j u 1 2 f n 2 u 2 log f e u dx in x h n u j u h j 1 n 1 2 and 2n is the area of the unit sphere s 2n 1 in r 2n in this paper we prove the inf u x jp u is always attained for p 2n the existence of minimizers of jp at the critical value p 2n is a delicate problem the proof depends on the blowup analysis for a sequence of bubbling solutions here we develop a local version of the method of moving planes to exclude the boundary bubbling the existence of minimizers for jp at the critical value p 2n is in contrast to the case of two dimensions

robustness of optimal intermittent search strategies in one two and three dimensions search problems at various scales involve a searcher be it a molecule before reaction or a foraging animal which performs an intermittent motion here we analyze a generic model based on such type of intermittent motion in which the searcher alternates phases of slow motion allowing detection and phases of fast motion without detection we present full and systematic results for different modeling hypotheses of the detection mechanism in space in one two and three dimensions our study completes and extends the results of our recent letter loverdo et al nat phys 4 134 2008 and gives the necessary calculation details in addition another modeling of the detection case is presented we show that the mean target detection time can be minimized as a function of the mean duration of each phase in one two and three dimensions importantly this optimal strategy does not depend on the details of the modeling of the slow detection phase which shows the robustness of our results we believe that this systematic analysis can be used as a basis to study quantitatively various real search problems involving intermittent behaviors

degree counting and shadow system for toda system of rank two one bubbling abstract we initiate the program for computing the leray schauder topological degree for toda systems of rank two this program still contains a lot of challenging problems for analysts as the first step we prove that if a sequence of solutions u 1 k u 2 k blows up then one of h j e u j k m h j e u j k d v g j 1 2 tends to a sum of dirac measures this is so called the phenomena of weak concentration our purposes in this article are i to introduce the shadow system due to the bubbling phenomena when one of parameters i crosses 4 and j 4 n where 1 i j 2 ii to show how to calculate the topological degree of toda systems by computing the topological degree of the general shadow systems iii to calculate the topological degree of the shadow system for one point blow up we believe that the degree counting formula for the shadow system would be useful in other problems

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

elephants can always remember exact long range memory effects in a non markovian random walk we consider a discrete time random walk where the random increment at time step t depends on the full history of the process we calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p at a critical value p 1 c 1 2 where memory effects vanish there is a transition from a weakly localized regime where the walker elephant returns to its starting point to an escape regime inside the escape regime there is a second critical value where the random walk becomes superdiffusive the probability distribution is shown to be governed by a non markovian fokker planck equation with hopping rates that depend both on time and on the starting position of the walk on large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time dependent spring constant k 2p 1 t the solution of this problem is a gaussian distribution with time dependent mean and variance which both depend on the initiation of the process

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

survival of an evasive prey abstract we study the survival of a prey that is hunted by n predators the predators perform independent random walks on a square lattice with v sites and start a direct chase whenever the prey appears within their sighting range the prey is caught when a predator jumps to the site occupied by the prey we analyze the efficacy of a lazy minimal effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range otherwise the prey stays still we show that if the sighting range of such a lazy prey is equal to 1 lattice spacing at least 3 predators are needed in order to catch the prey on a square lattice in this situation we establish a simple asymptotic relation lnp ev t n v 2lnp imm t between the survival probabilities of an evasive and an immobile prey hence when the density n v of the predators is low 1 the lazy evasion strategy leads to the spectacular increase of the survival probability we also argue that a short sighting prey its sighting range is smaller than the sighting range of the predators undergoes an effective superdiffusive motion as a result of its encounters with the predators whereas a far sighting prey performs a diffusive type motion

bubbling solutions for a skew symmetric chern simons system in a torus abstract we establish the existence of bubbling solutions for the following skew symmetric chern simons system u 1 1 e 2 e u 2 1 e u 1 4 i 1 n 1 p i 1 u 2 1 e 2 e u 1 1 e u 2 4 i 1 n 2 p i 2 over a parallelogram with doubly periodic boundary condition where e 0 is a coupling parameter and p denotes the dirac measure concentrated at p we obtain that if n 1 1 n 2 1 1 there exists an e 0 0 such that for any e 0 e 0 the above system admits a solution u 1 e u 2 e satisfying u 1 e and u 2 e blow up simultaneously at the point p and 1 e 2 e u j e 1 e u i e 4 n i p 1 i j 2 i j as e 0 where the location of the point p defined by 1 12 satisfies the condition 1 13

a single predator charging a herd of prey effects of self volume and predator prey decision making we study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape we explicitly consider the self volume of the prey restraining their dynamics on the lattice the movement of both predator and prey is chosen to include an intelligent decision making step based on their respective sighting ranges the radius in which they can detect the other species prey cannot recognise each other besides the self volume interaction after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase respectively we consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey we find that the prey s sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight we characterise the dynamics in terms of the mean distance between the predator and the nearest prey it turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple ornstein uhlenbeck picture reducing the many body problem to a simple two body problem by imagining predator and nearest prey to be connected by an effective hookean bond all features of the model such as prey density and sighting ranges merge into the effective binding constant

a generalized trudinger moser inequality on a compact riemannian surface abstract let g be a compact riemannian surface let h be two smooth functions on with d v g 0 and h 0 h 0 in this paper using a method of blow up analysis we prove that the functional 1 j h u 1 2 g u 2 d v g 8 1 d v g u d v g 8 log h e u d v g is bounded from below in w 1 2 g moreover we obtain a sufficient condition under which j h attains its infimum in w 1 2 g these results generalize the main results in ding jost li wang 1997 and yang zhu 2017

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

spatial fluctuations of a surviving particle in the trapping reaction we consider the trapping reaction a b b where a and b particles have a diffusive dynamics characterized by diffusion constants da and db the interaction with b particles can be formally incorporated in an effective dynamics for one a particle as was recently shown by bray et al 2003 phys rev e 67 060102 we use this method to compute in space dimension d 1 the asymptotic behaviour of the spatial fluctuation z2 t 1 2 for a surviving a particle in the perturbative regime da db 1 for the case of an initially uniform distribution of b particles we show that for t 1 z2 t 1 2 t with 1 4 by contrast the fluctuations of paths constrained to return to their starting point at time t grow with the larger exponent 1 3 numerical tests are consistent with these predictions

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

harmonic maps of the two sphere into a complex grassmann manifold ii where qt is a complex valued one form defined up to a factor of absolute value 1 this form qp defines a complex structure on m for x e m the space f x has an orthogonal space f x of dimension n k we denote by f x and f x their corresponding projective spaces of dimensions k 1 and n k 1