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

convergences for sequences of sets and linear mappings let e and f be nontrivial real normed vector spaces and e f be the class of continuous linear operators from e into f let a a n n n e f we shall suppose that a lim a n in a certain sense strong convergence uniform convergence

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

stable soliton resolution for exterior wave maps in all equivariance classes abstract in this paper we consider finite energy l equivariant wave maps from r t x 1 3 r b 0 1 s 3 with a dirichlet boundary condition at r 1 and for all l n each such l equivariant wave map has a fixed integer valued topological degree and in each degree class there is a unique harmonic map which minimizes the energy for maps of the same degree we prove that an arbitrary l equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class i e soliton resolution this extends the recent results of the first second and fourth authors on the 1 equivariant equation to higher equivariance classes and thus completely resolves a conjecture of bizon chmaj and maliborski who observed this asymptotic behavior numerically the proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d 2 l 3 which are established in the companion paper 13

a characterization of proper minimal points as solutions of sublinear optimization problems abstract proper minimal points are of special interest in multiple objective optimization problems in this paper proper minimal points of sets in locally convex spaces and proper minimal solutions of constrained vector problems in topological linear spaces are characterized as solutions of scalar optimization problems in the latter case no differentiability assumption is made

brownian motion on lie groups and open quantum systems we study the twirling semigroups of super operators namely certain quantum dynamical semigroups that are associated in a natural way with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group the link connecting this class of semigroups of operators with classical brownian motion is clarified it turns out that every twirling semigroup associated with a finite dimensional representation is a random unitary semigroup and conversely every random unitary semigroup arises as a twirling semigroup using standard tools of the theory of convolution semigroups of measures and of convex analysis we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite dimensional unitary representation of a lie group

radially symmetric wave maps from 1 2 dimensional minkowski space to the sphere by a blow up analysis as in 8 for a related problem we rule out concentration of energy for radially symmetric wave maps from the 1 2 dimensional minkowski space to the sphere when combined with the local existence and regularity results of christodoulou and tahvildar zadeh for this problem our result implies global existence of smooth solutions to the cauchy problem for radially symmetric wave maps for smooth radially symmetric data

global regularity of wave maps from r3 1 to surfaces we consider wave maps with smooth compactly supported initial data of small norm from r3 1 to certain 2 dimensional riemannian manifolds and show that they stay smooth globally in time our methods are based on the introduction of a global coulomb gauge as in 17 followed by a dynamic separation as in 8 we then rely on an adaptation of t tao s methods used in his recent breakthrough result 24

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

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

on the soliton resolution for equivariant wave maps to the sphere we consider finite energy corotational wave maps with target manifold s2 we prove that for a sequence of times they decompose as a sum of decoupled harmonic maps in the light cone and a smooth wave map in the blowup case or a linear scattering term in the global case up to an error which tends to 0 in the energy space 2015 wiley periodicals inc

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

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

two bubble dynamics for threshold solutions to the wave maps equation we consider the energy critical wave maps equation mathbb r 1 2 rightarrow mathbb s 2 in the equivariant case with equivariance degree k ge 2 it is known that initial data of energy 8 pi k and topological degree zero leads to global solutions that scatter in both time directions we consider the threshold case of energy 8 pi k we prove that the solution is defined for all time and either scatters in both time directions or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction in the latter case we describe the asymptotic behavior of the scales of the two harmonic maps the proof combines the classical concentration compactness techniques of kenig merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

uniqueness of topological solutions of self dual chern simons equation with collapsing vortices abstract we consider the following chern simons equation 0 1 u 1 e 2 e u 1 e u 4 i 1 n p i e in where is a 2 dimensional flat torus e 0 is a coupling parameter and p stands for the dirac measure concentrated at p in this paper we proved that the topological solutions of 0 1 are uniquely determined by the location of their vortices provided the coupling parameter e is small and the collapsing velocity of vortices p i e is slow enough or fast enough comparing with e this extends the uniqueness results of choe 5 and tarantello 22 meanwhile for any topological solution defined in r 2 whose linearized operator is non degenerate we construct a sequence of topological solutions u e of 0 1 whose asymptotic limit is exactly after rescaling around 0 a consequence is that non uniqueness of topological solutions in r 2 implies non uniqueness of topological solutions on torus with collapsing vortices

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

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

asymptotic stability of large energy harmonic maps under the wave map from 2d hyperbolic spaces to 2d hyperbolic spaces abstract in this paper we prove that the large energy harmonic maps from h 2 to h 2 are asymptotically stable under the wave map equation this work is the last paper in a series of works 46 45 44 which together prove the asymptotic stability for all admissible harmonic maps of arbitrary size without any symmetry assumptions

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

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

quantization of time like energy for wave maps into spheres in this article we consider large energy wave maps in dimension 2 1 as in the resolution of the threshold conjecture by sterbenz and tataru commun math phys 298 1 139 230 2010 commun math phys 298 1 231 264 2010 but more specifically into the unit euclidean sphere mathbb s n 1 subset mathbb r n with n geq2 and study further the dynamics of the sequence of wave maps that are obtained in sterbenz and tataru commun math phys 298 1 231 264 2010 at the final rescaling for a first finite or infinite time singularity we prove that on a suitably chosen sequence of time slices at this scaling there is a decomposition of the map up to an error with asymptotically vanishing energy into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm concentrating on the null boundary and converging to a constant locally in the interior of the cone in the energy space similar and stronger results have been recently obtained in the equivariant setting by several authors cote commun pure appl math 68 11 1946 2004 2015 cote commun pure appl math 69 4 609 612 2016 cote am j math 137 1 139 207 2015 cote et al am j math 137 1 209 250 2015 krieger commun math phys 250 3 507 580 2004 where better control on the dispersive term concentrating on the null boundary of the cone is provided and in some cases the asymptotic decomposition is shown to hold for all time here however we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to lin and riviere ann math 149 2 785 829 1999 duke math j 111 177 193 2002 which we make work here in the hyperbolic context of sterbenz and tataru commun math phys 298 1 231 264 2010

regular articleconvergences for sequences of sets and linear mappings let e f be normed vector spaces and let a an be continuous linear operators from e into f consider x xn y yn arbitrary nonempty closed subsets of e f we study the problem if x lim xn y lim yn in the sense of kuratowski or mosco or attouch wets do we have a x lim a xn a 1 y lim a 1n yn in the same sense in solving this problem we improve recent results obtained by i schochetman and r smith and published in this journal and extend some results of g beer established for the case f r

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

new type of nontopological bubbling solutions in the su 3 chern simons model in r2 abstract we are concerned with su 3 chern simons model in r 2 nontopological solutions were shown to exist in 10 in a radially symmetric situation also in 10 there constructed a sequence of partially bubbling solutions of which one component only bubbles we here construct a sequence of radial nontopological bubbling solutions of another type explicitly one component of these bubbling solutions locally converges to a nice function but both components blow up partially in different regions at infinity at the same time this type of bubbling solutions not only is a new phenomenon to this type of systems but also shows a more complicated pattern of the nontopological solution structure than the abelian chern simons higgs equation proposed by 13 15

asymptotic behavior of condensate solutions in the chern simons higgs theory we study the asymptotic behavior of condensate solutions in the chern simons higgs model as the chern simons coupling parameter tends to zero using the variational method we prove that there exist condensate solutions which show concentration phenomena

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

stability of efficient sets continuity of mobile polarities whenever for every x e d such that a x afi xo for i 1 2 n one hash x fi x for each i pareto optimization is also referred to as uecror multi objecfiue or multi criteria optimization multi objective maximization problems 0 1 have been nowadays extended to the situations where f is a mapping from x to a set y equipped with a transitive relation 2 this relation is frequently called a preference consider first the image a f d of the constraint d an element y of a is called 3 efficient up to indifference or in the broad sense if for every y e a y z y implies y 5 yo it is said to be strongly z eficient or efficient in the narrow sense if for each y e a y 2 y implies y y the two notions coincide for antisymmetric transitive relations we shall be primarily concerned with efficient points in the broad sense the set of efficient points of a with respect to 7 will be denoted by max a max z a and that of strongly efficient points by max a an element of y is called an efficient value of 0 1 if it is an efficient point of f d an element x0 of d is a solution of 0 1 if f xo is an efficient point of f d we shall use the latter s to denote the set of solutions of 0 1

gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space in this paper we study k equivariant wave maps from the hyperbolic plane into the 2 sphere as well as the energy critical equivariant su 2 s u 2 yang mills problem on 4 dimensional hyperbolic space the latter problem bears many similarities to a 2 equivariant wave map into a surface of revolution as in the case of 1 equivariant wave maps considered in 9 both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target then it is asymptotically stable in the energy space however for a stationary solution that covers a large enough portion of the target we prove that the schrodinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap as there is no a priori nonlinear obstruction to asymptotic stability this gives evidence for the existence of metastable states i e solutions with anomalously slow decay rates in these simple geometric models

generalized e quasi solutions in multiobjective optimization problems existence results and optimality conditions abstract this paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant we introduce the concept of generalized e quasi solution that extends other well known approximate efficiency notions of the literature and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets moreover we prove several existence results and a bound for the generalized e quasi solution set under convexity assumptions and by using asymptotic analysis tools finally we develop optimality conditions for a particular case of these kinds of e quasi solutions in nonsmooth convex and nonconvex problems

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

ill posedness for one dimensional wave maps at the critical regularity we show that the wave map equation in r 1 1 is in general ill posed in the critical space 1 2 and the besov space 1 2 1 2 the problem is attributed to the bad behavior of the one dimensional bilinear expression d 1 fdg in these spaces

absence of self similar blow up and local well posedness for the constant mean curvature wave equation abstract in this note we consider the constant mean curvature wave equation in 1 2 dimensions we show that it does not admit any self similar blow up we also remark that the equation is locally well posed for initial data in h 3 2

two dimensional euler flows with concentrated vorticities for a planar model of euler flows proposed by tur and yanovsky 2004 we construct a family of velocity fields w e for a fluid in a bounded region with concentrated vorticities e for e 0 small more precisely given a positive integer and a sufficiently small complex number a we find a family of stream functions e which solve the liouville equation with dirac mass source e e 2 e e 4 pa e in e 0 on for a suitable point p p a e the vorticities e e concentrate in the sense that 1 e 4 pa e 8 1 j 1 pa e a j 0 as e 0 j 1 where the satellites a 1 a 1 denote the complex 1 roots of a the point p a e lies close to a zero point of a vector field explicitly built upon derivatives of order 1 of the regular part of green s function of the domain

stability of stationary equivariant wave maps from the hyperbolic plane in this paper we initiate the study of equivariant wave maps from 2d hyperbolic space bbb h 2 into rotationally symmetric surfaces this problem exhibits markedly different phenomena than its euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain in particular when the target is bbb s 2 we find a family of equivariant harmonic maps bbb h 2 to bbb s 2 indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere these maps have energies taking all values between zero and the energy of the unique co rotational euclidean harmonic map q rm euc from bbb r 2 to bbb s 2 given by stereographic projection we prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere indeed we prove strichartz estimates for the operator obtained by linearizing around such a harmonic map however for harmonic maps with energies approaching the euclidean energy of q rm euc asymptotic stability via a perturbative argument based on strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator when the target is bbb h 2 we find a continuous family of asymptotically stable equivariant harmonic maps bbb h 2 to bbb h 2 with arbitrarily small and arbitrarily large energies this stands in sharp contrast to the corresponding problem on euclidean space where all finite energy solutions scatter to zero as time tends to infinity

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

profile decompositions for wave equations on hyperbolic space with applications the goal for this paper is twofold our first main objective is to develop bahouri gerard type profile decompositions for waves on hyperbolic space recently such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy with an eye towards further applications we develop this theory in a fairly general framework which includes the case of waves on hyperbolic space perturbed by a time independent potential our second objective is to use the profile decomposition to address a specific nonlinear problem namely the question of global well posedness and scattering for the defocusing energy critical semi linear wave equation on three dimensional hyperbolic space possibly perturbed by a repulsive time independent potential using the concentration compactness rigidity method introduced by kenig and merle we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as t rightarrow pm infty this proof serves as a blueprint for the arguments in the work 45 by the authors where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane

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

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

convergence analysis of solution sets for fuzzy optimization problems abstract in this paper we first consider fuzzy optimization problem in short fop and sequences of fuzzy optimization problems in short fop n then we establish the concept of d convergence of a sequence of fuzzy mappings and its characterizations finally the painleve kuratowski lower convergence and the painleve kuratowski convergence of approximate solution sets for these problems are proposed using d convergence of a sequence of fuzzy mappings some examples are provided to illustrate the essentialness of the imposed assumptions

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

the wave equation on hyperbolic spaces abstract we study the dispersive properties of the wave equation associated with the shifted laplace beltrami operator on real hyperbolic spaces and deduce new strichartz estimates for a large family of admissible pairs as an application we obtain local well posedness results for the nonlinear wave equation

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

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge

equivariant wave maps in two space dimensions singularities of corotational wave maps from 1 2 dimensional minkowski space into a surface n of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into in consequence for noncompact target surfaces of revolution the cauchy problem for wave maps is globally well posed 2003 wiley periodicals inc

well posedness and stability for abstract spline problems abstract in this work well posedness and stability properties of the abstract spline problem are studied in the framework of reflexive spaces tykhonov well posedness is proved without restrictive assumptions in the context of hilbert spaces also the stronger notion of levitin polyak well posedness is established a sequence of parametric problems converging to the given abstract spline problem is considered in order to study stability under natural assumptions convergence results for sequences of solutions of the perturbed problems are obtained

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

blow up at infinity in the su 3 chern simons model part i abstract we consider non topological solutions of a nonlinear elliptic system problem see 1 4 below derived from the s u 3 chern simons models in r 2 the existence of non topological solutions even for radial symmetric case has been a long standing open problem recently choe kim and lin in 7 8 showed the existence of radial symmetric non topological solution when the vortex points collapse however the arguments in 7 8 cannot work for an arbitrary configuration of vortex points in this paper we develop a new approach by using different scalings for different components of the system to construct a family of non topological solutions which blows up at infinity

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

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

variational inclusions for contingent derivative of set valued maps in this paper we give two versions of ky fan s inequality for set valued maps acting between normed vector spaces and we consider sufficient conditions to solve a variational inclusion problem concerning derivatives of set valued maps a selection result for set valued maps between finite dimensional vector spaces and its contingent derivative is obtained as well from this result we derive some conditions for the existence of a solution of a generalized variational inequality problem

convergence of minimal sets in convex vector optimization we study the behavior of the minimal sets of a sequence of convex sets left a n right converging to a given set a the main feature of the present work is the use of convexity properties of the sets a n and a to obtain upper and lower convergence of the minimal frontiers we emphasize that we study both kuratowski painleve convergence and attouch wets convergence of minimal sets moreover we prove stability results that hold in a normed linear space ordered by a general cone in order to deal with the most common spaces ordered by their natural nonnegative orthants e g c left left a b right right l p and l p left mathbb r right for 1 leq p leq infty we also make a comparison with the existing results related to the topics considered in our work

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 existence and stability of approximate solutions of perturbed vector equilibrium problems abstract in this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of painleve kuratowski to the initial set then we introduce the concept of approximate solution for equilibrium problem governed by set valued maps and we study the stability of these solutions the particular case of linear continuous operators is considered as well

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

global regularity of wave maps from r 2 1 to h 2 small energy we demonstrate that wave maps with smooth initial data and small energy from r2 1 to the lobatchevsky plane stay smooth globally in time our method is similar to the one employed in 18 however the multilinear estimates required are considerably more involved and present novel technical challenges in particular we shall have to work with a modification of the functional analytic framework used in 30 33 18

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

parametrized multicriteria optimization continuity and closedness of optimal multifunctions this paper studies continuity properties of the marginal multifunction and solution multifunction for optimization problems with respect to a preorder relation among others we present a number of results which are generalizations to multicriteria problems of theorems due to c berge w hogan t tanino and y sawaragi

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

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

agmon kato kuroda theorems for a large class of perturbations we prove asymptotic completeness for operators of the form h l on l2 rd d 2 where l is an admissible perturbation our class of admissible perturbations contains multiplication operators defined by real valued potentials v lq rd q d 2 d 1 2 if d 2 then we require q 1 3 2 as well as real valued potentials v satisfying a global kato condition the class of admissible perturbations also contains first order differential operators of the form a a for suitable vector potentials a our main technical statement is a new limiting absorption principle which we prove using techniques from harmonic analysis related to the stein tomas restriction theorem