the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

structure of boundary blow up solutions for quasi linear elliptic problems ii small and intermediate solutions the structure of positive boundary blow up solutions to quasi linear elliptic problems of the form pu f u u on 1 0 in z1 z2 large small and intermediate solutions are obtained for sufficiently large it is known from part i see structure of boundary blow up solutions for quasilinear elliptic problems part i large and small solutions preprint that the large solution is the unique large solution to the problem we will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions our results are new even for the case p 2

ap weights for nondoubling measures in rn and applications we study an analogue of the classical theory of ap weights in r n without assuming that the underlying measure is doubling then we obtain weighted norm inequalities for the centered hardy littlewood maximal function and corresponding weighted estimates for nonclassical calderon zygmund operators in the sense of ntv1 we also consider commutators of those calderon zygmund operators with bounded mean oscillation functions bmo extending the main result from crw finally we study self improving properties of poincare b m o type inequalities within this context more precisely we show that if f is a locally integrable function satisfying 1 q r q f fq d a q for all cubes q then it is possible to deduce higher l p integrability result of f assuming certain simple geometric condition on the functional a

calderon zygmund operators with non doubling measures in this survey the authors review some basic results on calderon zygmund theory on r d with the non doubling radon measure instead of the usual doubling condition the measure is only assumed to satisfy some growth condition that is there exist positive con stants n 0 d a ndc such that b x r cr n for all x r d and r 0 in this setting the results that we review are the following the hardy space h 1 and the regularized bmo space rbmo some interpolation results related to h 1 a nd rbmo the calderon zygmund decomposition the t 1 theorem and the properties of the calderon zygmund op erator on lebesgue spaces and hardy spaces cotlar s inequality and the boundedness of the maximal calderon zygmund operator the behaviors of the multilinear calderon zygmund op erator on lebesgue spaces weighted norm inequalities for the calderon zygmund operator the boundedness of the commutator generated by the calderon zygmund operator and rbmo function moreover some new progress obtained recently is also included

uniform asymptotic stability of non linear singularly perturbed general and large scale systems new general advantageous criteria are established for uniform asymptotic stability of singularly perturbed general and large scale systems possibly containing non differentiable non linearities which often appear in control systems for systems with multiple time scales caused by several small parameters or for large scale systems an essential order reduction of aggregation matrices is achieved the results are conceptually and numerically applied to the absolute stability analysis of singularly perturbed lurie postnikov systems

realization of prescribed patterns in the competition model we demonstrate that for any prescribed set of finitely many disjoint closed subdomains d1 dm of a given spatial domain in rn if d1 d2 a1 a2 c d e are positive continuous functions on and b x is identically zero on d d1 dm and positive in the rest of then for suitable choices of the parameters and all small e 0 the competition model ut x t d1 x u x t a1 x u e 1b x 1 u2 c x uv vt x t d2 x v x t a2 x v e x v2 e 1d x uv x t 0 under natural boundary conditions on possesses an asymptotically stable positive steady state solution ue ve that has pattern d that is roughly speaking as e 0 ue converges to a positive function over d while it converges to 0 over the rest of on the other hand ve converges to 0 over d but converges to some positive function in the rest of in other words the two competing species ue and ve become spatially segregated as e 0 with ue concentrating on d and ve concentrating on d

uniqueness of the solution of a semilinear boundary value problem on considere le probleme aux valeurs limites suivant 2 u f x u 0 x u x 0 x sur pour e 0 il y a un e 0 tel que pour 0 e e ce probleme ait une solution positive unique u dans c 2 c

uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain n via a generalisation of serrin s sweeping principle in the case when is an annulus it is shown that the solution is radially symmetric

effects of a degeneracy in the competition model part ii perturbation and dynamical behaviour this is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of in part i we mainly discussed the existence of two kinds of steady state solutions of this system namely the classical steady states and the generalized steady states here we use these solutions to determine the dynamics of the model we do this with the help of the perturbed model where b x is replaced by b x e which itself is a classical competition model this approach also reveals the interesting relationship between the steady state solutions both classical and generalized of the above system and that of the perturbed system

the semiadditivity of continuous analytic capacity and the inner boundary conjecture let a e be the continuous analytic capacity of a compact set e c c in this paper we obtain a characterization of a in terms of curvature of measures with zero linear density and we deduce that a is countably semiadditive this result has important consequences for the theory of uniform rational approximation on compact sets in particular it implies the so called inner boundary conjecture 1 introduction the continuous analytic capacity a was introduced by erokhin and vitushkin see vi in the 1950s in order to study problems of uniform rational approximation in compact subsets of the complex plane up to now the geometric properties of the capacity a have not been well understood it has been an open question if a is semiadditive as a set function that is to say if

boundary blow up solutions with a spike layer abstract we prove that for large 0 the boundary blow up problem u u u a 1 u in u has a solution with a spike located near the most centered part of our proof is based on the reduction method and uses a cut off technique to avoid the difficulties for the estimates near the boundary of

besov spaces with non doubling measures suppose that is a radon measure on r d which may be non doubling the only condition on is the growth condition namely there is a constant c 0 0 such that for all x supp and r 0 b x r c 0 r n where 0 0 is a real number which depends on the non doubling measure c 0 n and d the method used to define these spaces is new even for the classical case as applications the lifting properties of these spaces by using the riesz potential operators and the dual spaces are obtained

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth abstract asymptotic behaviour of small solutions of a class of singularly perturbed quasilinear elliptic dirichlet problems with critical and supercritical growth is studied as e 0