astala s conjecture on distortion of hausdorff measures under quasiconformal maps in the plane let e subset mathbb c be a compact set g mathbb c to mathbb c be a k quasiconformal map and let 0 t 2 let mathcal h t denote t dimensional hausdorff measure then mathcal h t e 0 quad rightarrow quad mathcal h t left ge right 0 quad t frac 2kt 2 left k 1 right t

improved bounds for the overall properties of a nonlinear composite dielectric the construction of generalized hashin shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material in recent work a nonlinear comparison medium has been used however this requires detailed knowledge of the properties of the trial fields that are employed the fields used have the property of bounded mean oscillation and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled some recent results concerning riesz transforms and the beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds in addition an exact solution is established for a particular class of composites

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

on astala s theorem for martingales and fourier multipliers abstract we exhibit a large class of symbols m on r d d 2 for which the corresponding fourier multipliers t m satisfy the following inequality if d e are measurable subsets of r d with e d and d then d e t m e x d x e e ln d 2 e if e d 2 d e 1 2 d e ln e d e if e d 2 here denotes the lebesgue measure on i r d when d 2 these multipliers include the real and imaginary parts of the beurling ahlfors operator b and hence the inequality is also valid for b with the right hand side multiplied by 2 the inequality is sharp for the real and imaginary parts of b this work is motivated by k astala s celebrated results on the gehring reich conjecture concerning the distortion of area by quasiconformal maps the proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales these martingale bounds are of interest on their own right

quasiconformal distortion of riesz capacities and hausdorff measures in the plane in this paper we prove the sharp distortion estimates for the quasi conformal mappings in the plane both in terms of the riesz capacities from non linear potential theory and in terms of the hausdorff measures

a note on the strong maximum principle and the compact support principle abstract in this note we are concerned with the strong maximum principle smp and the compact support principle csp for non negative solutions to quasilinear elliptic inequalities of the form div a u u g u f u 0 in and div a u u g u f u 0 in r n b r 0 respectively we give new conditions on the data a g f to obtain smp and csp when these conditions are particularized to the m laplacian and pure power nonlinearities we completely classify the data according to the validity of the csp or the smp in doing so we clarify the general situation and we consider a case not covered in the literature

existence of positive solutions for nonlinear elliptic equations with convection terms abstract we prove the existence of positive solutions for the equation i 1 n x i a i x u u f x u u in under the dirichlet boundary condition where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient u we develop an approach based on approximate solutions and on a new strong maximum principle

pucci s conjecture and the alexandrov inequality for elliptic pdes in the plane the inequality of alexandrov bakel man and pucci is a basic tool in the theory of linear elliptic partial differential equations pdes which are not in divergence form as well as in the more general theory of nonlinear elliptic pdes here in two dimensions we prove the sharp form of the maximum principle as conjectured by pucci in 1966 give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting these results make use of the substantial recent advances in the planar theory of quasiconformal mappings astala was supported in part by the academy of finland projects 34082 and 41933 iwaniec by the national science foundation grants dms 0301582 and dms 0244297 martin by the n z marsden fund and the n z royal society james cook fellowship ams 1991 classification primary 30c60 35j15 35j70

burkholder integrals morrey s problem and quasiconformal mappings inspired by morrey s problem on rank one convex functionals and the burkholder integrals of his martingale theory we find that the burkholder functionals bp p 2 are quasiconcave when tested on deformations of identity f id c with bp df x 0 pointwise or equivalently deformations such that df 6 p p 2jf in particular this holds in explicit neighbourhoods of the identity map among the many immediate consequences this gives the strongest possible l estimates for the gradient of a principal solution to the beltrami equation fz z fz for any p in the critical interval 2 6 p 6 1 1 f examples of local maxima lacking symmetry manifest the intricate nature of the problem

teichmuller s problem in space abstract quasiconformal homeomorphisms of the whole space r n onto itself normalized at one or two points are studied in particular the stability theory the case when the maximal dilatation tends to 1 is in the focus our main result provides a spatial analogue of a classical result due to teichmuller unlike teichmuller s result our bounds are explicit explicit bounds are based on two sharp well known distortion results the quasiconformal schwarz lemma and the bound for linear dilatation moreover bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations finally we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of r n 0 onto itself

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

integral estimates of conformal derivatives and spectral properties of the neumann laplacian abstract in this paper we study integral estimates of derivatives of conformal mappings d of the unit disc d c onto bounded domains that satisfy the ahlfors condition these integral estimates lead to estimates of constants in sobolev poincare inequalities and by the rayleigh quotient we obtain spectral estimates of the neumann laplace operator in non lipschitz domains quasidiscs in terms of the quasi conformal geometry of the domains specifically the lower estimates of the first non trivial eigenvalues of the neumann laplace operator in some fractal type domains snowflakes were obtained

sobolev regular flows of non lipschitz vector fields abstract we show that vector fields with exponentially integrable derivatives admit a well defined flow of homeomorphisms x t w 1 p t l o c for some p t 1 at least for small times when the field is certain riesz potential of a bounded function the result becomes global in time due to techniques from geometric function theory the local result also applies to the flows arising from yudovich solutions to the planar euler system with bounded vorticity

distributional solutions of the beltrami equation abstract we study the distributional solutions to the generalized beltrami equation under sobolev assumptions on the beltrami coefficients in this setting we prove that these distributional solutions are true quasiregular maps and they are smoother than expected that is they have second order derivatives in l l o c 1 e for some e 0

the geometry of the weil petersson metric in complex dynamics in this work we study an analogue of the weil petersson metric on the space of blaschke products of degree 2 proposed by mcmullen we show that the weilpetersson metric is incomplete and study its metric completion our work parallels known results for the teichmuller space of a punctured torus

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

l p bounds for the beurling ahlfors transform let b denote the beurling ahlfors transform defined on l p c 1 p 00 the celebrated conjecture of t iwaniec states that its l p norm b p p 1 1 where p max p p p 1 in this paper the new upper estimate b p 1 575 p 1 1 p is found

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

a finite step algorithm via a bimatrix game to a single controller non zero sum stochastic game given a non zero sum discounted stochastic game with finitely many states and actions one can form a bimatrix game whose pure strategies are the pure stationary strategies of the players and whose penalty payoffs consist of the total discounted costs over all states at any pure stationary pair it is shown that any nash equilibrium point of this bimatrix game can be used to find a nash equilibrium point of the stochastic game whenever the law of motion is controlled by one player the theorem is extended to undiscounted stochastic games with irreducible transitions when the law of motion is controlled by one player examples are worked out to illustrate the algorithm proposed

a martingale study of the beurling ahlfors transform inrn abstract we apply the theory of martingale transforms to study the beurling ahlfors transform s in dimensions n 2 this operator reveals a rich structure through its representation as a martingale and we obtain new results concerning the operator norm of s acting on the class of differential forms having l p coefficients in particular we show that its norm is independent of the dimension when restricted to k forms and we present new essen type norm inequalities related to this martingale structure finally we suggest a purely analytic method to further investigate these norms which up to now has been lacking

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

vertical linear complementarity and discounted zero sum stochastic games with arat structure in this paper we consider a two person zero sum discounted stochastic game with arat structure and formulate the problem of computing a pair of pure optimal stationary strategies and the corresponding value vector of such a game as a vertical linear complementarity problem we show that cottle dantzig s algorithm a generalization of lemke s algorithm can solve this problem under a mild assumption

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

harmonic measure on locally flat domains we will review work with tatiana toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation

hausdorff measure of quasicircles abstract s smirnov 2010 10 proved recently that the hausdorff dimension of any k quasicircle is at most 1 k 2 where k k 1 k 1 in this paper we show that if is such a quasicircle then h 1 k 2 b x r c k r 1 k 2 for all x c r 0 where h s stands for the s hausdorff measure on a related note we derive a sharp weak integrability of the derivative of the riemann map of a quasidisk

sign and area in nodal geometry of laplace eigenfunctions the paper deals with asymptotic nodal geometry for the laplace beltrami operator on closed surfaces given an eigenfunction f corresponding to a large eigenvalue we study local asymmetry of the distribution of sign f with respect to the surface area it is measured as follows take any disc centered at the nodal line f 0 and pick at random a point in this disc what is the probability that the function assumes a positive value at the chosen point we show that this quantity may decay logarithmically as the eigenvalue goes to infinity but never faster than that in other words only a mild local asymmetry may appear the proof combines methods due to donnelly fefferman and nadirashvili with a new result on harmonic functions in the unit disc

on almost fuchsian manifolds almost fuchsian manifold is a class of complete hyperbolic threemanifolds a such three manifold is a quasi fuchsian manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of 1 1 in such a manifold the minimal surface is unique and embedded hence one can parametrize these hyperbolic three manifolds by their minimal surfaces in this paper we obtain estimates on several geometric and analytical quantities of an almost fuchsian manifold m in terms of the data on the minimal surface in particular we obtain an upper bound for the hyperbolic volume of the convex core of m and an upper bound on the hausdorff dimension of the limit set associated to m we also constructed a quasi fuchsian manifold which admits more than one minimal surface and it does not admit a foliation of closed surfaces of constant mean curvature

the generalized linear complementarity problem revisited given a vertical block matrixa we consider in this paper the generalized linear complementarity problem vlcp q a introduced by cottle and dantzig we formulate this problem as a linear complementarity problem with a square matrixm a formulation which is different from a similar formulation given earlier by lemke our formulation helps in extending many well known results in linear complementarity to the generalized linear complementarity problem we also show that the class of vertical block matrices which cottle and dantzig s algorithm can process is the same as the class of equivalent square matrices which lemke s algorithm can process we also present some degree theoretic results on a vertical block matrix

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

multiple existence results of solutions for the neumann problems via super and sub solutions by variational methods we provide existence results of multiple solutions for quasilinear elliptic equations under the neumann boundary condition our main result shows the existence of two constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the unknown function the studied equations contain the p laplacian problems as a special case moreover we give a result concerning a local minimizer in c1 versus w1 p auxiliary results of independent interest are also obtained a density property for the space w1 p a strong maximum principle of zhang s type and a moser s iteration scheme depending on a parameter

existence and uniqueness results for double phase problems with convection term abstract in this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient abstract the goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the dirichlet boundary condition and a convection multivalued term an existence theorem for positive solutions of the problem is established by applying the method of subsolution supersolution together with truncation and comparison techniques

discrepancy densities for planar and hyperbolic zero packing we study the problem of geometric zero packing recently introduced by hedenmalm 7 there are two natural densities associated with this problem the discrepancy density h given by h liminfr 1

fuzzy approximation to an infeasible generalized linear complementarity problem in this paper we use fuzzy approximation to deal with infeasibility in generalized linear complementarity problem the solution of such problem with some degree of satisfaction is obtained by finding the minimum relaxation which should be allowed in the original problem

non stretch mappings for a sharp estimate of the beurling ahlfors operator abstract in this paper we identify certain classes of non stretch mappings that enjoy a sharp estimate of the beurling ahlfors operator we first make use of a property of subharmonic functions to prove that the banuelos wang conjecture and the iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate beltrami equation by utilizing the principal solutions of beltrami equations we further explicitly construct some classes of non stretch mappings for which the banuelos wang conjecture and the iwaniec conjecture are true

dimension of quasicircles we introduce canonical antisymmetric quasiconformal maps which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle as an application we prove astala s conjecture that the hausdorff dimension of a k quasicircle is at most 1 k 2

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

existence results for nonlinear elliptic equations with leray lions operator and dependence on the gradient the existence of positive solutions for nonlinear elliptic problems under dirichlet boundary condition is studied as well as the compactness and directness of the solution set the main novelties consist in the presence of a leray lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution our approach relies on the method of sub supersolution nonlinear regularity theory and strong maximum principle

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

mappings of bmo bounded distortion this paper can be viewed as a sequel to the work 9 where the theory of mappings of bmo bounded distortion is developed largely in even dimensions using singular integral operators and recent developments in the theory of higher integrability of jacobians in hardy orlicz spaces in this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions the planar case was studied earlier by g david 4 in particular he obtained existence theorems modulus of continuity estimates and bounds on area distortion for mappings of bmo distortion in fact in slightly more generality we obtain similar results in all even dimensions one of our main new results here is the extension of the classical theorem of painleve concerning removable singularties for bounded analytic functions to the class of mappings of bmo bounded distortion the setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem or the measurable riemann mapping theorem which is not available in higher dimensions we give a construction to show our results are qualitatively optimal another surprising fact is that there are domains which support no bounded quasiregular mappings but admit

quasiconformal mappings as a tool to study certain two dimensional g closure problems a recent theorem due to astala establishes the best exponent for the area distortion of planar k quasiconformal mappings we use a refinement of astala s theorem due to eremenko and hamilton to prove new bounds on the effective conductivity of two dimensional composites the bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions for n 2 we prove the optimality of the bounds under certain additional assumptions on the g closure parameters

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

continuity of capacity of a holomorphic motion abstract we study the behavior of various set functions under holomorphic motions we show that under such deformations logarithmic capacity varies continuously while analytic capacity may not

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

on a quasilinear elliptic problem with convection term and nonlinear boundary condition abstract the first part of this paper deals with existence of solutions to the quasilinear elliptic problem p div a x u f x u u in a x u g x u u p 2 u on involving a general nonhomogeneous differential operator namely div a and caratheodory functions f r r n r and g r r under appropriate conditions on the perturbations we show that p possesses a bounded solution in the second part we consider the special case when div a is the p q laplacian with a parameter 0 and study the asymptotic behavior of solutions as goes to zero or to infinity a uniqueness result is also provided

bloch functions asymptotic variance and geometric zero packing motivated by a problem in quasiconformal mapping we introduce a problem in complex analysis with its roots in the mathematical physics of the bose einstein condensates in superconductivity the p

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

quasiconformal mappings and sharp estimates for the distance to l in some function spaces abstract we provide several estimates which involve the distance to l in some function spaces the composition operator induced by a quasiconformal mapping and the logarithm of the jacobian of a quasiconformal mapping our results are sharp in the two dimensional case

snowballs are quasiballs we introduce snowballs which are compact sets in r 3 homeomorphic to the unit ball they are 3 dimensional analogs of domains in the plane bounded by snowflake curves for each snowball b a quasiconformal map f colon r 3 to r 3 is constructed that maps b to the unit ball

on the first eigenvalues of free vibrating membranes in conformal regular domains in 1961 g polya published a paper about the eigenvalues of vibrating membranes the free vibrating membrane corresponds to the neumann laplace operator in bounded plane domains in this paper we obtain estimates for the first non trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains this class includes convex domains john domains etc on the basis of our estimates we conjecture that the eigenvalues of the neumann laplace operator depend on the hyperbolic metrics of plane domains we propose a new method for the estimates which is based on weighted poincare sobolev inequalities obtained by the authors recently

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have