self dual instantons and holomorphic curves a gradient flow of a morse function on a compact riemannian manifold is said to be of morse smale type if the stable and unstable manifolds of any two critical points intersect transversally for such a morse smale gradient flow there is a chain complex generated by the critical points and graded by the morse index the boundary operator has as its x y entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the morse indices is 1 the homology of this chain complex agrees with the homology of the underlying manifold m and this can be used to prove the morse inequalities cf 33 26 around 1986 floer generalized this idea to infinite dimensional variational problems in which every critical point has infinite morse index but the moduli spaces of connecting orbits form finite dimensional manifolds for every pair of critical points the dimensions of these spaces give rise to a relative morse index and the boundary operator is defined by counting connecting

pointwise gradient estimates for a class of singular quasilinear equations with measure data abstract local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data div a x u in a bounded and possibly nonsmooth domain in r n here div a x u is modeled after the p laplacian our results extend earlier known results to the singular case in which 3 n 2 2 n 1 p 2 1 n

periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

finite dimensional irreducible representations of the quantum algebra uq c2 in this article the communtation relations of the generators of quantum algebra uq c2 are analyzed and irreducible q tensor operators of rank 1 2 of quantum algebra suq 2 are constructed by means of the property of a q tensor operator finite irreducible representations of uq c2 can be easily obtained some low dimensional representations are tabulated

connected floer homology of covering involutions using the covering involution on the double branched cover of s 3 branched along a knot and adapting ideas of hendricks manolescu and hendricks hom lidman we define new knot concordance invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots

pointwise gradient estimates abstract we survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear possibly degenerate elliptic and parabolic equations

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

a global regularity result for some degenerate elliptic systems abstract we establish a caccioppoli type inequality for a second order degenerate elliptic systems of p laplacian type as a direct consequence exploiting classical campanato s approach and the hole filling technique due to widman we are able to prove a global regularity result on morrey s and l q spaces with q p

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

representations of uq sl n at roots of unity the gelfand zetlin basis for representations of uq sl n is improved to better fit the case when q is a root of unity the usual q deformed representations as well as the nilpotent periodic cyclic semi periodic semi cyclic and also some atypical representations are now described with the same formalism

self duality for the two component asymmetric simple exclusion process we study a two component asymmetric simple exclusion process asep that is equivalent to the asep with second class particles we prove self duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra uq x1d524 x1d529 3 we construct all invariant measures in explicit form and discuss some of their properties we also prove a sum rule for the duality functions

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

gradient estimates via the wolff potentials for a class of quasilinear elliptic equations abstract in this paper we obtain the pointwise gradient estimates via the nonlinear wolff potentials for weak solutions of a class of non homogeneous quasilinear elliptic equations with measure data

on definite strongly quasipositive links and l space branched covers abstract we investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised seifert forms and apply our results to the problem of determining when such a link can have an l space cyclic branched cover in particular we show that if n 1 2 n 1 is the dual garside element and b n k p b n is a strongly quasipositive braid whose braid closure b is definite then k 2 implies that b is one of the torus links t 2 q t 3 4 t 3 5 or pretzel links p 2 2 m p 2 3 4 applying 7 theorem 1 1 we deduce that if one of the standard cyclic branched covers of b is an l space then b is one of these links we show by example that there are strongly quasipositive braids n p whose closures are definite but not one of these torus or pretzel links we also determine the family of definite strongly quasipositive 3 braids and show that their closures coincide with the family of strongly quasipositive 3 braids with an l space branched cover

riesz potentials and nonlinear parabolic equations the spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric riesz potential of the right hand side datum exactly as in the case of the heat equation heat kernels type estimates persist in the nonlinear case

links of rational singularities l spaces and lo fundamental groups we prove that the link of a complex normal surface singularity is an l space if and only if the singularity is rational this via a result of hanselman et al taut foliations on graph manifolds 2015 arxiv 1508 0591 proving the conjecture of boyer et al math ann 356 4 1213 1245 2013 shows that a singularity link is not rational if and only if its fundamental group is left orderable if and only if it admits a coorientable taut foliation

universal potential estimates abstract we prove a class of endpoint pointwise estimates for solutions to quasilinear possibly degenerate elliptic equations in terms of linear and nonlinear potentials of wolff type of the source term such estimates allow to bound size and oscillations of solutions and their gradients pointwise and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients in particular local estimates in holder lipschitz morrey and fractional spaces as well as calderon zygmund estimates follow as a corollary in a unified way moreover estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied the classical wolff potential estimate by kilpelainen mal and trudinger wang as well as recent wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases

gradient estimates via non linear potentials we present pointwise gradient bounds for solutions to p laplacean type non homogeneous equations employing non linear wolff type potentials and then prove similar bounds via suitable caloric potentials for solutions to parabolic equations

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

on l spaces and left orderable fundamental groups examples suggest that there is a correspondence between l spaces and three manifolds whose fundamental groups cannot be left ordered in this paper we establish the equivalence of these conditions for several large classes of manifolds in particular we prove that they are equivalent for any closed connected orientable geometric three manifold that is non hyperbolic a family which includes all closed connected orientable seifert fibred spaces we also show that they are equivalent for the twofold branched covers of non split alternating links to do this we prove that the fundamental group of the twofold branched cover of an alternating link is left orderable if and only if it is a trivial link with two or more components we also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in text homeo s 1

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

weighted gradient estimates for the parabolic p laplacian equations abstract in this paper we obtain the weighted l q estimates of the gradients of weak solutions for the quasilinear parabolic equations of p laplacian type with small bmo coefficients for every e 0 q 1 if u p l w l o c 1 e t and f p l w l o c q t then we prove that u p l w l o c q t our results improve the known results for such equations using a harmonic analysis free technique

irreducible highest weight representations of quantum groupsu q gl n c explicit recurrence formulas of canonical realization boson representation for quantum enveloping algebrasu q gl n c are given using them irreducible highest weight representations ofu q gl n c are obtained as restriction of representation of fock space to invariant subspace generated by vacuum as a cyclic vector

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

realisations of the real semisimple lie algebras a method of construction for an arbitrary real semisimple lie algebra g a method is presented of constructing skew hermitian realisations in terms of quantum canonical variables pi qi the construction starts with a decomposition g n b g0b n b of g which is a simple generalisation of the triangle decomposition employing substantially an induced representation of g with respect to a suitable representation sigma of the subalgebra g0b n b it is shown that the realisations obtained are schurean as examples the realisations for the algebras sp 2 r and gl 3 r are calculated explicitly

potential estimates of superquadratic elliptic systems with vmo coefficients in reifenberg domains abstract the purpose of this paper is to study an inhomogeneous dirichlet problem of a superquadratic nonlinear elliptic system with vmo coefficients in reifenberg flat domains we derive pointwise potential estimates for weak solution to the dirichlet problem as a consequence we derive global integrability estimates for solutions with respect to lorentz and morrey norms

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

an end point orlicz type estimate for nonlinear elliptic equations abstract we study nonlinear elliptic equations of p laplacian type in divergence form to establish a natural calderon zygmund type theory of an orlicz space type where the lebesgue space is the special case with t t q p f p l 1 q 2 r d u p l 1 q r in the previous results the estimates obtained were strictly above the natural exponents and the function such as t t log 1 t was ruled out for the candidate of but with our approach can be selected up to the end point case of the estimates and the functions t t and t t log 1 t can be additionally selected for

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

singular quasilinear and hessian equations and inequalities we solve the existence problem in the renormalized or viscosity sense and obtain global pointwise estimates of solutions for quasilinear and hessian equations with measure coefficients and data including the following model problems pu uq fk u uq u 0 on rn or on a bounded domain rn here p is the p laplacian defined by pu div u u p 2 and fk u is the k hessian i e the sum of the k k principal minors of the hessian matrix d2u k 1 2 n and are general nonnegative measurable functions or measures on

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

the q analog of the boson algebra its representation on the fock space and applications to the quantum group in this paper a realization of the q deformed boson operators on the fock space from a generally algebraic point of view is given the representations of the quantum group cn q are thereby constructed in terms of this realization some infinite and finite dimensional representations of the q analog of the heisenberg weyl algebra are obtained on certain quotient spaces finally the q deformed differential realization of quantum group given by alvarez gaume gomez and sierra preprint cern th 5369 89 is derived from the boson realization

regularity estimates for nonlinear elliptic measure data problems with nonstandard growth abstract we prove a global calderon zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right hand side is a bounded radon measure minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate

nonlinear parabolic problems with lower order terms and related integral estimates abstract we deal with the solutions to nonlinear parabolic equations of the form u t div a x t d u g x t u f x t on t t 0 under standard growth conditions on g and a with f only assumed to be integrable to the power 1 we prove general local decay estimates for level sets of the solutions u and the gradient d u which imply very general estimates in rearrangement function spaces lebesgue orlicz lorentz and non rearrangement ones up to lorentz morrey spaces

rotation numbers of products of circle homeomorphisms let h homeo s 1 be the group of orientation preserving homeomorphisms of the circle our main question is the following for given numbers 71 and y2 what are the possible rotation numbers of a product qlq2 of elements ql q2 of h with rotation numbers 71 y2 what if some or all of ql q2 qlq2 are required to be conjugate to rotations our original motivation was the question of which seifert fibered 3 manifolds admit transverse foliations which we discuss in sect 7 the answers turn out to be much more subtle than we originally expected we can make the question more precise by working in the universal covering group 7 of h this is the group of homeomorphisms of f which lift from an orientation preserving homeomorphism of s 1 r z that is 7 q r r i q monotonically increasing q r 1 q r 1 for all r r 7 is simply connected since it is a convex subset of ir for y define sh h by

global lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators abstract this article is a contribution to the study of regularity theory for nonlinear elliptic equations the aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as div u p 2 u a x u q 2 u div f p 2 f a x f q 2 f which arises from double phase functional problems in particular the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators this work extends that of colombo and mingione 2016 23 and byun and oh 2017 9 by dealing with the global estimates in lorentz spaces it also extends our recent result tran and nguyen 2020 66 concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators for future research the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces

gradient potential estimates in non linear elliptic obstacle problems with measure data abstract in this paper we derive pointwise estimates for the gradients of solutions to elliptic obstacle problems of p laplace type with measure data moreover we address a borderline problem in the regularity theory and give a sharp criterion for continuity of the gradient of a solution other applications are lorentz space estimates for the gradient and a criterion for c 1 regularity of solutions

on the set of l space surgeries for links abstract we study the qualitative structure of the set ls of integral l space surgery slopes for links with two components it is known that the set of l space surgery slopes for a nontrivial l space knot is always a positive half line however already for two component torus links the set ls has a very complicated structure e g in some cases it can be unbounded from below in order to probe the geometry of this set we ask if it is bounded from below for more general l space links with two components for algebraic two component links we provide three complete characterizations for the boundedness from below one in terms of the alexander polynomial one in terms of the embedded resolution graph and one in terms of the so called h function introduced by the authors in 9 it turns out that ls is bounded from below for most algebraic links if it is unbounded from below it must contain a negative half line parallel to one of the axes we also give a sufficient condition for boundedness for arbitrary l space links

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

unitary representations of the q oscillator algebra we classify the unitary irreducible representations of the q oscillator algebra bb q2b b 1 by the sign of the definite operator k bb b b k 0 corresponds to fock representations and k 0 to non fock ones k 0 being a degenerate case we present their link to representations of extended q oscillator algebra aa qa a q n obtained earlier we give examples of all representations besides standard ones we describe a new non fock representation of the macfarlane type

riesz potential type estimates for nonlinear elliptic equations abstract for p 2 we study nonlinear elliptic equations of p laplacian type with measure data div a d u u x in when the nonlinearity depends on the variable of the possibly unbounded solution itself we obtain the point wise gradient bound and the gradient continuity in terms of a certain riesz potential of the right hand side measure

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

a new method of determining an irreducible representation of quantum slq 3 algebra i the explicit forms of the irreducible representation matrices for the quantum slq 3 enveloping algebra are computed by a new technique

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

good lambda and muckenhoupt wheeden type bounds in quasilinear measure datum problems with applications weighted good lambda type inequalities and muckenhoupt wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data such results are obtained globally over sufficiently flat domains in mathbb r n in the sense of reifenberg the principal operator here is modeled after the p laplacian where for the first time singular case frac 3n 2 2n 1 p le 2 frac 1 n is considered those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data as an application sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear riccati type equation having a gradient source term with linear or super linear power growth

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

good λ type bounds of quasilinear elliptic equations for the singular case abstract in this paper we study the good type bounds for renormalized solutions to nonlinear elliptic problem d i v a x u in u 0 on where r n is a finite radon measure and a is a monotone carathedory vector valued function defined on w 0 1 p the operator a satisfies growth and monotonicity conditions and the p capacity uniform thickness condition is imposed on r n for the singular case 3 n 2 2 n 1 p 2 1 n in fact the same good type estimates were also studied by quoc hung nguyen and nguyen cong phuc for instance in nguyen and phuc 0000 and nguyen 0000 authors method was also confined to the case of 3 n 2 2 n 1 p 2 1 n but under the assumption of is the reifenberg flat domain and the coefficients of a have small bmo bounded mean oscillation semi norms otherwise the same problem was considered in phuc 2014 in the regular case of p 2 1 n in this paper we extend their results taking into account the case 3 n 2 2 n 1 p 2 1 n and without the hypothesis of reifenberg flat domain on and small bmo semi norms of a moreover in rest of this paper we also give the proof of the boundedness property of maximal function on lorentz spaces and also the global gradient estimates of solution