the sharp weighted bound for general calderon zygmund operators for a general calderon zygmund operator t on r n it is shown that ktfkl2 w c t sup q a q w q w 1 a k fkl2 w for all muckenhoupt weights w 2 a2 this optimal estimate was known as the a2 conjecture a recent result of perez treil volberg reduced the problem to a testing condition on indicator functions which is verified in this paper the proof consists of the following elements i a variant of the nazarov treil volberg method of random dyadic systems with just one random system and completely without bad parts ii a resulting representation of a general calderon zygmund operator as an average of dyadic shifts and iii improvements of the lacey petermichl reguera estimates for these dyadic shifts which allow summing up the series in the obtained representation

non homogeneous t1 theorem for bi parameter singular integrals abstract we prove a non homogeneous t 1 theorem for certain bi parameter singular integral operators moreover we discuss the related non homogeneous journe s lemma and product bmo theory

existence nonexistence and asymptotic behavior of boundary blow up solutions to p x laplacian problems with singular coefficient abstract this paper investigates the problem p x u x f x u 0 in u x as d x 0 where p x u d i v u p x 2 u is called the p x laplacian and x is a singular coefficient the existence and nonexistence of boundary blow up solutions is discussed and the asymptotic behavior of boundary blow up solutions is given in particular we do not assume radial symmetric conditions and the pointwise different exact blow up rate of solutions has been discussed

large solutions of semilinear elliptic equations under the keller osserman condition abstract we consider the equation u p x f u where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on 0 satisfies f 0 0 f s 0 for s 0 and the keller osserman condition 1 f s 1 2 d s where f s 0 s f t d t we establish conditions on the function p that are necessary and sufficient for the existence of positive solutions bounded and unbounded of the given equation

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

a remark on the radial and nonradial solutions for an elliptic system with a quadratic gradient term recently the authors yang and zhang in y yang x zhang entire blow up solutions of semilinear elliptic system with quadratic gradient term abstract and applied analysis 2012 have been obtained sufficiently conditions for the existence and non existence of radially symmetric solutions for the system of equations du i u i 2 a i x f i u 1 u d on r n n 3 with i 1 d and d 1 2 3 4 our aim in this paper is to give a necessary and a sufficient condition but for the existence of large or explosive solutions for the same class of systems moreover we intend to obtain a non radial solution for the considered problem we also point out that our framework in the obtained results improve and complete some of the results obtained by yang and zhang

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

existence of entire explosive positive solutions of quasi linear elliptic equations in this paper our main purpose is to establish the existence of entire explosive positive radial solutions for quasi linear elliptic equations the main results of the paper extend previously known results both for the super linear and sub linear cases

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

some new results on the existence of bounded positive entire solutions for quasilinear elliptic equations in this paper our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations under new conditions the main results of the present paper are new and extend the previously known results

existence of large solutions for a class of quasilinear elliptic equations abstract we consider the quasilinear elliptic equation div u p 2 u a x f u u 0 in where p 1 a x 0 a x 0 a x c and f is continuous and non decreasing on 0 satisfies f 0 0 f s 0 for s 0 and the keller osserman condition 1 f s 1 p d s f s 0 s f t d t we establish conditions on the function a that are necessary and sufficient for the existence of positive solutions bounded and unbounded of the given equation

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

blow up boundary solutions of semilinear elliptic problems we consider the following semilinear elliptic equation u p x f u in u 0 u 0 in 1 where rn n 3 is a smooth domain bounded or possibly unbounded with compact possibly empty boundary we assume throughout this paper that p is a non negative function such that p c0 if is bounded and p c loc otherwise the nonlinearity f is assumed to ful ll f 1 f c1 0 f 0 f 0 0 and f 0 on 0 and the keller osserman condition see 7 13

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

large and entire large solutions for a class of nonlinear problems abstract in this paper our main purpose is to establish the existence of entire large positive solutions to the quasilinear equation p u x h u we give also necessary conditions for the existence and nonexistence of such solutions we distinguish the cases when is radial and when it is nonradial

existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems abstract we show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div u m 2 u p x g v div v n 2 v q x f u on r n where f and g are positive and non decreasing functions on 0 satisfying the keller osserman condition

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

existence of positive bounded entire solutions for quasilinear elliptic equations in this paper our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations the main results of the present paper are new and extend the previously known results

existence of explosive positive solutions of quasilinear elliptic equations abstract in this paper our main purpose is to consider the quasilinear equation div u p 2 u m x f u on a domain r n n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and m is a nonnegative continuous function with the property that any zero of m is contained in a bounded domain in such that m is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists for un boundary including r n we show that a similar result holds where u x as x within and u x as x

on optimally scaled systems for second order scalar singularity perturbed problems the conditioning of the singularly perturbed scalar dirichlet problem is considered it is shown how this is related to the conditioning of an appropriate associated first order system through this the dichotomy of the solution space a concept that only makes sense in a vectorial setting can be investigated two typical equations are studied in more detail one with possible boundary layers on both sides of the interval and one with an internal layer i e the turning point case the results are applied to obtain estimates of global discretization errors for difference methods several examples illustrate the analysis

representation of bi parameter singular integrals by dyadic operators abstract we prove a dyadic representation theorem for bi parameter singular integrals that is we represent certain bi parameter operators as averages of rapidly decaying sums of what we call bi parameter shifts a new version of the product space t 1 theorem is established as a consequence

some obstacles in characterising the boundedness of bi parameter singular integrals the famous t1 theorem for classical calderon zygmund operators is a characterisation for their boundedness in l 2 in the bi parameter case on the other hand the current t1 theorem is merely a collection of sufficient conditions this difference in mind we study a particular dyadic bi parameter singular integral operator namely the full mixed bi parameter paraproductp which is precisely the operator responsible for the outstanding problems in the bi parameter theory we make several remarks about p the common theme of which is to demonstrate the delicacy of the problem of finding a completely satisfactory product t1 theorem for example p need not be unconditionally bounded if it is conditionally bounded a major difference compared to the corresponding one parameter model operators moreover currently the theory even lacks a characterisation for the potentially easier unconditional boundedness the product bmo condition is sufficient but far from necessary we show by example that unconditional boundedness does not even imply the weaker rectangular bmo condition