global regularity for a class of quasi linear local and nonlocal elliptic equations on extension domains abstract we consider the solvability of quasi linear elliptic equations with either local or nonlocal neumann robin or wentzell boundary conditions defined in the generalized sense on a bounded w 1 p extension domain whose boundary is an upper d set for an appropriate d 0 then we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions known for bounded lipschitz domains to bounded w 1 p extension domains whose boundaries are upper d sets by showing that such weak solutions are globally holder continuous consequently we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type neumann robin or wentzell may be uniformly continuous up to the boundary

traces of sobolev functions on fractal type sets and characterization of extension domains abstract we describe traces of sobolev functionsu w1 p r n 1 r nin terms of sobolev spaces on metric spaces 7 our results apply to smooth submanifolds fractal subsets as well as to open subsets of r n in particular if 0 r nis a john domain then we characterize thosew1 p functions which can be extended tow1 p r n if is uniform then this result implies jones extension theorem 14 in the case of traces on fractal subsets our results are related to those of jonsson and wallin 16

besov spaces on closed subsets of ⁿ motivated by the need in boundary value problems for partial differential equations classical trace theorems characterize the trace to a subset f of rn of sobolev spaces and besov spaces consisting of functions defined on rn if f is a linear subvariety rd of rn or a d dimensional smooth submanifold of rn this was generalized in 2 to the case when f is a ddimensional fractal set of a certain type in this paper traces are described when f is an arbitrary closed set the result may also be looked upon as a whitney extension theorem in lp

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

on quasisymmetry of quasiconformal mappings abstract suppose that f d d is a quasiconformal mapping where d and d are domains in r n and that d is a broad domain we show that for every arcwise connected subset a in d the weak quasisymmetry of the restriction f a a f a implies its quasisymmetry as a consequence we see that the answer to one of the open problems raised by heinonen from 1989 is affirmative under the additional condition that a is arcwise connected

sobolev lp2 functions on closed subsets of r2 abstract for each p 2 we give intrinsic characterizations of the restriction of the homogeneous sobolev space l p 2 r 2 to an arbitrary finite subset e of r 2 the trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the menger curvature of triangles with vertices in e

sobolev extension by linear operators let l r be the sobolev space of functions with mth derivatives lying in l r assume that n p for e r let l e denote the space of restrictions to e of functions in l r we show that there exists a bounded linear map t lm p e l r such that for any f l e we have tf f on e we also give a formula for the order of magnitude of f lm p e for a given f e r when e is finite

elliptic equations and maps of bounded length distortion on considere l equation elliptique d ordre 2 l u i f 1 n 1 a ij ju 0 ou les coefficients a ij sont des fonctions c 1 dans un domaine d de r n

the quasiconformal subinvariance property of john domains in mathbb r n and its applications the main aim of this paper is to give a complete solution to one of the open problems raised by heinonen from 1989 concerning the subinvariance of john domains under quasiconformal mappings in mathbb r n as application the quasisymmetry of quasiconformal mappings is discussed

an euler type method for the strong approximation of the cox ingersoll ross process we analyse the strong approximation of the cox ingersoll ross cir process in the regime where the process does not hit zero by a positivity preserving drift implicit euler type method as an error criterion we use the p th mean of the maximum distance between the cir process and its approximation on a finite time interval we show that under mild assumptions on the parameters of the cir process the proposed method attains up to a logarithmic term the convergence of order 1 2 this agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with lipschitz coefficients despite the fact that the cir process has a non lipschitz diffusion coefficient

signal detection in fractional gaussian noise several results related to the reproducing kernel hilbert space of fractional brownian motion are presented to facilitate the study of signal detection in additive fractional gaussian noise this hilbert space is completely characterized and an alternative characterization for the restriction of this class of functions to a compact interval 0 t is given infinite and finite interval whitening filters for fractional brownian motion are also developed the application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional gaussian noise a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed

elliptic equations with bmo nonlinearity in reifenberg domains given p 2 8 we obtain the global w1 p estimate for the weak solution of a boundary value problem for an elliptic equation with bmo nonlinearity in a reifenberg domain assuming that the nonlinearity has sufficiently small bmo seminorm and that the boundary of the domain is sufficiently flat

sharp inequalities for maximal functions associated with general measures sharp weak type 1 1 and l p estimates in dimension one are obtained for uncentred maximal functions associated with borel measures which do not necessarily satisfy a doubling condition in higher dimensions uncentred maximal functions fail to satisfy such estimates analogous results for centred maximal functions are given in all dimensions

parabolic equations in time dependent reifenberg domains in this paper we define time dependent parabolic reifenberg domains and study lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains the basic assumption is that the principal coefficients are of parabolic bmo space with small parabolic bmo seminorms it is shown that lp estimates hold for time dependent parabolic d reifenberg domains

extensions and imbeddings we establish a connection between the sobolev imbedding theorem and the extendability of sobolev functions as applications we give geometric criteria for extendability and give a result on the dependence of the extension property on the exponentp

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

a bounded linear extension operator for l2 p r2 let l2 p r2 be the sobolev space of real valued functions on the plane whose hessian belongs to lp for any finite subset e r2 and p 2 let l2 p r2 e be the space of real valued functions on e equipped with the trace seminorm in this paper we construct a bounded linear extension operator t l2 p r2 e l2 p r2 we also provide an explicit formula that approximates the l2 p r2 e trace seminorm

bilipschitz homogeneous jordan curves we characterize bilipschitz homogeneous jordan curves by utilizing quasihomogeneous parameterizations we verify that rectifiable bilipschitz homogeneous jordan curves satisfy a chordarc condition we exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower hausdorff density zero we examine homeomorphisms between jordan curves

quasihyperbolic metric and quasisymmetric mappings in metric spaces in this paper we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces meanwhile we also show that quasi invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal at the end of this paper as an application of above theorems we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping

quasi linear variable exponent boundary value problems with wentzell robin and wentzell boundary conditions let p c0 1 be such that 1 p p 1 p p let rn rn be a bounded w1 p w1 p extension domain and let be an upper d ahlfors measure supported on with d n p n d n p n we investigate the solvability of a class of quasi linear boundary value problems involving the p p laplace and p p laplace beltrami operators and either classical wentzell robin boundary conditions p ud u p 2 u dhn 1 u p 2ud 0on or general fully wentzell type boundary conditions of the form p ud p ud u p 2 u dhn 1 u p 2ud 0on where l d l d is such that infx x 0 for some constant 0 0 0 0 and u dhn 1 denotes the generalized p p normal derivative on in the interpretative sense we prove that the realization of the p p laplace operator with both of the above boundary conditions generate nonlinear ultracontractive submarkovian c0c0 semigroups on l2 dx l2 d l2 dx l2 d and hence their associated first order cauchy problems are both well posed on lq dx lq d lq dx lq d for all measurable function q with 1 q q 1 q q in addition we investigate the associated quasi linear elliptic problem with general wentzell boundary conditions and obtain existence uniqueness and global regularity of weak solutions to this equation

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

whitney type extension theorems for jets generated by sobolev functions abstract let l p m r n 1 p be the homogeneous sobolev space and let e r n be a closed set for each p n and each non negative integer m we give an intrinsic characterization of the restrictions d f e m to e of m jets generated by functions f l p m 1 r n our trace criterion is expressed in terms of variations of corresponding taylor remainders of m jets evaluated on a certain family of well separated two point subsets of e for p this result coincides with the classical whitney glaeser extension theorem for m jets our approach is based on a representation of the sobolev space l p m 1 r n p n as a union of c m d r n spaces where d belongs to a family of metrics on r n with certain nice properties here c m d r n is the space of c m functions on r n whose partial derivatives of order m are lipschitz functions with respect to d this enables us to show that for every non negative integer m and every p n the very same classical linear whitney extension operator as in 31 provides an almost optimal extension of m jets generated by l p m 1 functions

sobolev w1p spaces on closed subsets of rn abstract for each p n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the sobolev space w p 1 r n to an arbitrary closed subset of r n

sobolev inequalities in arbitrary domains abstract a theory of sobolev inequalities in arbitrary open sets in r n is established boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order the relevant sobolev inequalities exhibit the same critical exponents as in the classical framework moreover they involve constants independent of the geometry of the domain and hence yield genuinely new results even in the case when just smooth domains are considered our approach relies upon new representation formulas for sobolev functions and on ensuing pointwise estimates which hold in any open set

trace operator and the dirichlet problem for elliptic equations on arbitrary bounded open sets abstract we consider the dirichlet problem and the weak dirichlet problem on a general possibly nonregular bounded domain for elliptic linear equation with uniformly elliptic divergence form operator we investigate carefully the relationship between weak integral probabilistic soft and the perron wiener brelot solutions of the problem we also study the relationship between solutions of the dirichlet problem and weak dirichlet problem to this end we extend the usual notion of the trace operator and consider the dirichlet problem with martin boundary

transparent boundary conditions for the helmholtz equation in some ramified domains with a fractal boundary the paper addresses a class of boundary value problems in some self similar ramified domains with the laplace or helmholtz equations much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains a self similar finite element method is proposed and tested it can be used for numerically computing the spectrum of the laplace operator with neumann boundary conditions as well as the eigenmodes the eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out finally a numerical method for the wave equation is addressed

an alternative coarse space for irregular subdomains and an overlapping schwarz algorithm for scalar elliptic problems in the plane in earlier work on domain decomposition methods for elliptic problems in the plane an assumption that each subdomain is triangular or a union of a few coarse triangles has often been made this is similar to what is required in geometric multigrid theory and is unrealistic if the subdomains are produced by a mesh partitioner in an earlier paper coauthored with axel klawonn the authors introduced a coarse subspace for an overlapping schwarz method with one degree of freedom for each subdomain vertex and one for each subdomain edge a condition number bound proportional to 1 log h h 2 1 h delta was established assuming only that the subdomains are john domains here h delta measures the relative overlap between neighboring subdomains and h h the maximum number of elements across individual subdomains we were also able to relate the rate of convergence to a parameter in an isoperimetric inequality for the subdomains into which the domain of the problem has been partitioned in this paper t

on planar sobolev lpm extension domains abstract for each m 1 and p 2 we characterize bounded simply connected sobolev l p m extension domains r 2 our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in an important geometrical ingredient for obtaining these results is a new square separation theorem it states that under certain natural assumptions on the relative positions of a point x and a square s there exists a similar square q which touches s and has the property that x and s belong to distinct connected components of q

on sobolev extension domains in r n abstract we describe a class of sobolev w p k extension domains r n determined by a certain inner subhyperbolic metric in this enables us to characterize finitely connected sobolev w p 1 extension domains in r 2 for each p 2

on the boundary values of sobolev wp1 functions abstract for each p n we use local oscillations to give intrinsic characterizations of the trace of the sobolev space w p 1 to the boundary of an arbitrary domain r n

hajlasz sobolev imbedding and extension abstract the author establishes some geometric criteria for a hajlasz sobolev m ball s p extension resp m ball s p imbedding domain of r n with n 2 s 0 1 and p n s resp p n s in particular the author proves that a bounded finitely connected planar domain is a weak cigar domain with 0 1 if and only if f p s r 2 m ball s p for some all s 1 and p 2 s where f p s r 2 denotes the restriction of the triebel lizorkin space f p s r 2 on

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

whitney s problem on extendability of functions and an intrinsic metric abstract we consider the space of functions with bounded k 1 th derivatives in a general domain in r n is every such function extendible to a function of the same class defined on the whole r n h whitney showed that the equivalence of the intrinsic geodesic metric in this domain to the euclidean one is sufficient for such extendability there was an old conjecture going back to h whitney that this equivalence is also necessary for extendability we disprove this conjecture and construct examples of domains in r 2 such that the above extendability holds but the analogous property for smaller k fails our study is based on a duality approach

conformality and q harmonicity in carnot groups we show that if f is a 1 quasiconformal map defined on an open subset of a carnot group g then composition with f preserves q harmonic functions we combine this with a regularity theorem for q harmonic functions and an algebraic regularity theorem for maps between carnot groups to show that f is smooth we give some applications to the study of rigidity

global estimates for quasilinear elliptic equations on reifenberg flat domains new global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems the coefficients are assumed to have small bmo seminorms and the boundary of the domain is sufficiently flat in the sense of reifenberg the main regularity estimates obtained are in weighted lorentz spaces other regularity results in lorentz morrey morrey and holder spaces are shown to follow from the main estimates