weighted gradient estimates for elliptic problems with neumann boundary conditions in lipschitz and semi convex domains abstract let n 2 and be a bounded lipschitz domain in r n in this article the authors investigate global weighted norm estimates for the gradient of solutions to neumann boundary value problems of second order elliptic equations of divergence form with real valued bounded measurable coefficients in more precisely for any given p 2 two necessary and sufficient conditions for w 1 p estimates of solutions to neumann boundary value problems respectively in terms of a weak reverse holder inequality with exponent p or weighted w 1 q estimates of solutions with q 2 p and some muckenhoupt weights are obtained as applications for any given p 1 and a p r n the class of muckenhoupt weights the authors establish weighted w 1 p estimates for solutions to neumann boundary value problems of second order elliptic equations of divergence form with small bmo coefficients on bounded semi convex domains as further applications the global gradient estimates are obtained respectively in weighted lorentz spaces lorentz morrey spaces weighted orlicz spaces and variable lebesgue spaces

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

computing upper and lower bounds in interval decision trees this article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities in tree models involving uncertainty expressed as intervals and or relations it is necessary for the evaluation to compute the upper and lower bounds of the expected values already in its simplest form computing a maximum of expectancies leads to quadratic programming qp problems unfortunately standard optimization methods based on qp and blp bilinear programming are too slow for the evaluation of decision trees in computer tools with interactive response times needless to say the problems with computational complexity are even more emphasized in multi linear programming mlp problems arising from multi level decision trees since standard techniques are not particularly useful for these purposes other non standard algorithms must be used the algorithms presented here enable user interaction in decision tools and are equally applicable to all multi linear programming problems sharing the same structure as a decision tree

global regularity for divergence form elliptic equations on quasiconvex domains abstract in this paper we introduce a notion of quasiconvex domain and show that the global w 1 p regularity holds on such domains for a wide class of divergence form elliptic equations the modified vitali covering lemma compactness method and the maximal function technique are the main analytical tools

global gradient estimates for the borderline case of double phase problems with bmo coefficients in nonsmooth domains abstract we consider a double phase problem with bmo coefficient in divergence form on a bounded nonsmooth domain the problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position which describes a feature of strongly anisotropic materials we obtain the global calderon zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small bmo and the boundary of the domain is sufficiently flat in the reifenberg sense

quantitative stability for the first dirichlet eigenvalue in reifenberg flat domains in rn abstract in this paper we prove that if and are close enough for the complementary hausdorff distance and their boundaries satisfy some geometrical and topological conditions then 1 1 c n where 1 resp 1 is the first dirichlet eigenvalue of the laplacian in resp and is the lebesgue measure of the symmetric difference here the constant 1 could be taken arbitrary close to 1 but strictly less and c is a constant depending on a lot of parameters including dimension n and some geometric properties of the domains

random reordering in sor type methods when iteratively solving linear systems by b by b with hermitian positive semi definite b and in particular when solving least squares problems for ax b ax b by reformulating them as aa y b aa y b it is often observed that sor type methods gau seidel kaczmarz perform suboptimally for the given equation ordering and that random reordering improves the situation on average this paper is an attempt to provide some additional theoretical support for this phenomenon we show error bounds for two randomized versions called shuffled and preshuffled sor that improve asymptotically upon the best known bounds for sor with cyclic ordering our results are based on studying the behavior of the triangular truncation of hermitian matrices with respect to their permutations

a linearly convergent doubly stochastic gauss seidel algorithm for solving linear equations and a certain class of over parameterized optimization problems consider the classical problem of solving a general linear system of equations ax b it is well known that the successively over relaxed gauss seidel scheme and many of its variants may not converge when a is neither diagonally dominant nor symmetric positive definite can we have a linearly convergent g s type algorithm that works for anya in this paper we answer this question affirmatively by proposing a doubly stochastic g s algorithm that is provably linearly convergent in the mean square error sense for any feasible linear system of equations the key in the algorithm design is to introduce a nonuniform double stochastic scheme for picking the equation and the variable in each update step as well as a stepsize rule these techniques also generalize to certain iterative alternating projection algorithms for solving the linear feasibility problem a x le b with an arbitrary a as well as high dimensional minimization problems for training over parameterized models in machine learning our results demonstrate that a carefully designed randomization scheme can make an otherwise divergent g s algorithm converge

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

weak maximum principle for biharmonic equations in quasiconvex lipschitz domains abstract in dimension two or three the weak maximum principle for biharmonic equation is valid in any bounded lipschitz domains in higher dimensions greater than three it was only known that the weak maximum principle holds in convex domains or c 1 domains and may fail in general lipschitz domains in this paper we prove the weak maximum principle in higher dimensions in quasiconvex lipschitz domains which is a sharp condition in some sense and recovers both convex and c 1 domains

weighted orlicz estimates for general nonlinear parabolic equations over nonsmooth domains we introduce a parabolic analogue of muckenhoupt weights to study optimal weighted regularity in orlicz spaces for a general nonlinear parabolic problem of p laplacian type in divergence form over a nonsmooth domain assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation bmo with respect to the spatial variables that the lateral boundary of the parabolic cylinder is reifenberg flat and that the associated weight belongs to a suitable parabolic muckenhoupt class we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted orlicz space our results extend the existing regularity estimates in lebesgue spaces to weighted orlicz spaces

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

elliptic equations with bmo coefficients in reifenberg domains the inhomogeneous dirichlet problems concerning divergence form elliptic equations are studied optimal regularity requirements on the coefficients and domains for the w1 p theory 1 p are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms the domain is a reifenberg domain these conditions for the w1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains in fact these domains might have fractal dimensions 2004 wiley periodicals inc

the calderon zygmund property for quasilinear divergence form equations over reifenberg flat domains abstract the results by palagachev 2009 3 regarding global holder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient moreover the principal coefficients are discontinuous with discontinuity measured in terms of small b m o norms and the underlying domain is supposed to have fractal boundary satisfying a condition of reifenberg flatness the results are extended to the case of parabolic operators as well

global regularity for divergence form elliptic equations in orlicz spaces on quasiconvex domains abstract in this paper we derive the global regularity of weak solutions to divergence form elliptic equations in the orlicz space w 0 1 l with 2 2 the domain considered here is quasiconvex which is a generalization of the reifenberg flat domain

global holder continuity of weak solutions to quasilinear divergence form elliptic equations abstract we derive global holder regularity for the w 0 1 2 weak solutions to the quasilinear uniformly elliptic equation div a i j x u d j u a i x u a x u d u 0 over a c 1 smooth domain r n n 2 the nonlinear terms are all of caratheodory type with coefficients a i j x u belonging to the class vmo of functions with vanishing mean oscillation with respect to x while a i x u and a x u d u exhibit controlled growths with respect to u and the gradient du

geometric analysis on cantor sets and trees dusing uniformization cantor type sets can be regarded as boundaries of rooted trees in this setting we show that the trace of a first order sobolev space on the boundary of a regular rooted tre

calderon zygmund theory for nonlinear elliptic problems with irregular obstacles abstract we consider a nonhomogeneous elliptic problem with an irregular obstacle involving a discontinuous nonlinearity over an irregular domain in divergence form of p laplacian type to establish the global calderon zygmund estimate by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term under the bmo smallness of the nonlinearity and sufficient flatness of the boundary in the reifenberg sense

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

on the equivalence between sor type methods for linear systems and the discrete gradient methods for gradient systems abstract inspired by the iterative nature of many discretization methods for continuous dynamical systems connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems have been studied since 1970s for stationary iterative methods solving linear systems chu 1988 2008 discussed a connection to continuous dynamical systems by using the explicit euler method however further understanding of stationary iterative methods might be limited due to the use of the explicit euler method this paper presents a new connection based on the so called discrete gradient methods between sor type methods and gradient systems there the key of the discussion is the equivalence between sor type methods and the discrete gradient methods applied to gradient systems the discussion leads to new interpretations for sor type methods for example a new derivation of sor type methods is found these methods monotonically decrease a certain quadratic function and a new interpretation of the relaxation parameter is obtained besides while studying the new connection a new discrete gradient is also obtained

brownian motion penetrating fractals an application of the trace theorem of besov spaces abstract for a closed connected set f in rn assume that there is a local regular dirichlet form a symmetric diffusion process on f whose domain is included in a lipschitz space or a besov space on f under some condition for the order of the space and the newtonian 1 capacity of f we prove that there exists a symmetric diffusion process on rn which moves like the process on f and like brownian motion on rn outside f as an application we will show that when f is a nested fractal or a sierpinski carpet whose hausdorff dimension is greater than n 2 we can construct brownian motion penetrating the fractal for the proof we apply the technique developed in the theory of besov spaces

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems

quasiconformal and bi lipschitz homeomorphisms uniform domains and the quasihyperbolic metric let d be a proper subdomain of r and kd the quasihyperbolic metric defined by the conformal metric tensor ds2 dist x dd 2ds2 the geodesies for this and related metrics are shown by purely geometric methods to exist and have lipschitz continuous first derivatives this is sharp for kd we also obtain sharp estimates for the euclidean curvature of such geodesies we then use these results to prove a general decomposition theorem for uniform domains in r in terms of embeddings of bi lipschitz balls we also construct a counterexample to the higher dimensional analogue of the decomposition theorem of gehring and osgood the defining density dist x 3d 1 need not be differentiable we primarily are interested in studying the geometry of the quasihyperbolic meric and its geodesies we show that the geodesies of this metric are c11 i e the arclength parametrisation has lipschitz continuous derivatives we show that this is best possible and obtain a sharp result on the euclidean curvature of such geodesies these results are proved for a more general class of metrics in particular metrics defined by locally lipschitz densities we then use these results to prove a gener alisation of the decomposition theorem of gehring and osgood for uniform domains in ri space i wish to express my sincere thanks to f w gehring for suggesting many of these problems for many helpful ideas and simplifications throughout and for allowing me to present his proof of theorem 3 7 i also wish to thank him and d herron for carefully reading the manuscript

quasilinear elliptic equations with vmo coefficients strong solvability and uniqueness in sobolev space w2 cl are proved for the dirichlet problem a j x u djju b x u du 0 a e i2 u p on d l it is assumed that the coefficients of the quasilinear elliptic operator satisfy caratheodory s condition the a i s are vmo functions with respect to x and structure conditions on b are required the main results are derived by means of the aleksandrov pucci maximum principle and leray schauder s fixed point theorem via a priori estimate for the 7 2 norm of the gradient

the conormal derivative problem for elliptic equations with bmo coefficients on reifenberg flat domains we study the inhomogeneous conormal derivative problem for the divergence form elliptic equation assuming that the principal coefficients belong to the bmo space with small bmo semi norms and that the boundary is delta reifenberg flat these conditions for the w 1 p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domain in fact the reifenberg flatness is the minimal regularity condition for the w 1 p theory

gradient estimates for elliptic systems in non smooth domains we obtain the global w 1 p 1 p estimate for the weak solution of an elliptic system with discontinuous coefficients in non smooth domains without using maximal function approach it is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale and that the tensor coefficients belong to bmo space with their bmo semi norms sufficiently small

applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients the paper concerns dirichlet s problem for second order quasilinear non divergence form elliptic equations with discontinuous coefficients we start with suitable structure growth and regularity conditions ensuring solvability of the problem under consideration fixing then a solution u 0 such that the linearized at u 0 problem is non degenerate we apply the implicit function theorem as a result we get that for all small perturbations of the coefficients there exists exactly one solution u u 0 which depends smoothly in w 2 p with p larger than the space dimension on the data for that no structure and growth conditions are needed and the perturbations of the coefficients can be general l functions of the space variable x moreover we show that the newton iteration procedure can be applied in order to obtain a sequence of approximate in w 2 p solutions for u 0

higher order elliptic operators of divergence form in c1 or lipschitz domains abstract we consider a 2 m th order elliptic operator of divergence form in a domain of r n whose leading coefficients are uniformly continuous in the paper y miyazaki the l p theory of divergence form elliptic operators under the dirichlet condition j differential equations 215 2005 320 356 we developed the l p theory including the construction of l p resolvents assuming that the boundary of is of class c m 1 the purpose of this paper is to show that the l p theory also holds when is a c 1 domain applying the inequalities of hardy type for the sobolev spaces

gradient estimates for nonlinear elliptic equations with vanishing neumann data in quasiconvex domains abstract we study the conormal derivative problem for an elliptic equation of p laplacian type with discontinuous coefficients in a non smooth domain in order to look for the minimal assumptions necessary to have the nonlinear calderon zygmund theory for such problem under the assumptions that the nonlinear operator is sufficiently close to the p laplacian operator in bmo semi norm and the boundary of the domain can be locally approximated by the convex boundary we prove that both the gradient and the associated nonhomogeneous term belong to the same l q space for every q p as far as the domain is concerned our regularity assumption on the boundary is weaker than any other one reported in this direction

weighted and regularity estimates for nonlinear equations on reifenberg flat domains abstract global weighted l p estimates are obtained for the gradient of solutions to nonlinear elliptic dirichlet boundary value problems over a bounded nonsmooth domain morrey and holder regularity of solutions are also established as a consequence these results generalize various existing estimates for nonlinear equations the nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation the boundary of the domain on the other hand may exhibit roughness but assumed to be sufficiently flat in the sense of reifenberg our approach uses maximal function estimates and vitali covering lemma and also known regularity results of solutions to nonlinear homogeneous equations

free boundary regularity for harmonic measures and poisson kernels one of the basic aims of this paper is to study the relationship between the geometry of hypersurface like subsets of euclidean space and the properties of the measures they support in this context we show that certain doubling properties of a measure determine the geometry of its support a radon measure is said to be doubling with constant c if c times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center we prove that if the doubling constant of a measure on r n 1 is close to the doubling constant of the ndimensional lebesgue measure then its support is well approximated by ndimensional a ne spaces provided that the support is relatively at to start with primarily we consider sets which are boundaries of domains in r n 1 the n dimensional hausdorfi measure may not be deflned on the boundary of a domain in r n 1 thus we turn our attention to the harmonic measure which is well behaved under minor assumptions see section 3 we obtain a new characterization of locally at domains in terms of the doubling properties of their harmonic measure see section 3 along these lines we investigate how the weak regularity of the poisson kernel of a domain determines the geometry of its boundary sections 5 and 6 pursue this goal as in alt and cafiarelli s work see ac c1 c2 and also jerison s j in both cases the goal is to prove that under the appropriate technical conditions at at points of the boundary the oscillation of the poisson kernel controls the oscillation of the unit normal vector the difierence between our work and the work in ac is that we measure the oscillation in an integral sense bmo estimates

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

continuity properties of the karush kuhn tucker point set in quadratic programming problems we obtain necessary and sufficient conditions for the set of the karush kuhn tucker points in a canonical quadratic programming problem to be upper semicontinuous or lower semicontinuous with respect to the problem parameters

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

trace theorems for vector fields in the setting of carnot caratheodory spaces we prove some trace theorems for sobolev functions we consider the trace on a non characteristic surface for hormander vector fields of step r ge 1 and the trace on the boundary of a class of domains in the grushin plane

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

elliptic equations with measurable nonlinearities in nonsmooth domains abstract we study elliptic equations with measurable nonlinearities in nonsmooth domains we establish an optimal global w 1 p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small bmo semi norm in the other variables while the underlying domain is sufficiently flat in the reifenberg sense that the boundary of the domain is locally trapped between two narrow strips

on solution stability of the linear complementarity problem in the paper 5 c d ha introduced the notion of stability of a linear complementarity problem at a solution point and established several sufficient conditions for stability to hold in the present paper we derive some new stability results which significantly improve ha s results included in these new results are i a complete characterization of the stability condition under a nonsingularity assumption and ii further characterizations of solution stability in the case of a linear complementarity problem with a fully semimonotone matrix

l p estimates for parabolic equations in reifenberg domains abstract the inhomogeneous neumann problem for parabolic equations in divergence form is studied an optimal regularity requirement on the domain for the l p theory is investigated assuming that the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms and that the domain is a reifenberg flat domain

embeddings of variable hajlasz sobolev spaces into holder spaces of variable order pointwise estimates in variable exponent sobolev spaces on quasi metric measure spaces are investigated based on such estimates sobolev embeddings into holder spaces with variable order are obtained this extends some known results to the variable exponent setting

on the continuity of the solution map in linear complementarity problems the continuity properties of the solution map mathcal s m q mapsto mathcal s m q are investigated where mathcal s m q denotes the solution set corresponding to the linear complementarily problem lcp m q a robinson type upper semicontinuity result is established for mathcal s and a generalization of the mangasarian shiau result concerning the lipschitzian property of mathcal s in the q variable is proved it is also shown that when the matrix is positive semidefinite or more generally a mathbf g matrix the solution map is lipschitz continuous with respect to the q vector if and only if the matrix is a mathbf p matrix

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

aw1 p estimate for solutions to mixed boundary value problems for second order elliptic differential equations on demontre que sous des hypotheses plutot faibles une solution d un probleme aux valeurs limites mixte pour une equation elliptique d ordre 2 est dans l espace de sobolev w 1 p pour p 2

a note on hajlasz sobolev spaces on fractals abstract we introduce hajlasz sobolev spaces involving walk dimensions on subsets of r n that will generally be fractals the relationship between this kind of new space and the besov space of jonsson and wallin is investigated the compact embedding theorems for hajlasz sobolev spaces are obtained

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

robustness and nondegenerateness for linear complementarity problems in this paper the main focus is on a stability concept for solutions of a linear complementarity problem a solution of such a problem is robust if it is stable against slight perturbations of the data of the problem relations are investigated between the robustness the nondegenerateness and the isolatedness of solutions it turns out that an isolated nondegenerate solution is robust and also that a robust nondegenerate solution is isolated since the class of linear complementarity problems with only robust solutions or only nondegenerate solutions is not an open set attention is paid to garcia s classgn of linear complementarity problems the nondegenerate problems ingn form an open set

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

on the stability of solutions to quadratic programming problems we consider the parametric programming problem q p of minimizing the quadratic function f x p x t ax b t x subject to the constraint cx d where x n a n n b n c m n d m and p a b c d is the parameter here the matrix a is not assumed to be positive semidefinite the set of the global minimizers and the set of the local minimizers to q p are denoted by m p and m loc p respectively it is proved that if the point to set mapping m loc is lower semicontinuous at p then m loc p is a nonempty set which consists of at most m n points where m n binom m text min m 2 n is the maximal cardinality of the antichains of distinct subsets of 1 2 m which have at most n elements it is proved also that the lower semicontinuity of m at p implies that m p is a singleton under some regularity assumption these necessary conditions become the sufficient ones

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

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

fourth order parabolic equations with weak bmo coefficients in reifenberg domains abstract we study optimal w 2 p regularity for fourth order parabolic equations with discontinuous coefficients in general domains we obtain the global w 2 p regularity for each 1 p under the assumption that the coefficients have suitably small bmo semi norm of weak type and the boundary of the domain is reifenberg flat the situation of our main theorem arises when the conductivity on fractals is controlled by a random variable in the time direction

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

lorentz estimate with a variable power for parabolic obstacle problems with non standard growths abstract we prove a global lorentz estimate for the variable power of the gradients of weak solution to parabolic obstacle problems with p t x growth over a bounded nonsmooth domain it is mainly assumed that the variable exponents p t x satisfy a strong type log holder continuity the associated nonlinearities are merely measurable in the time variable and have small bmo semi norms in the spatial variables while the underlying domain is quasiconvex

regularity theory in orlicz spaces for elliptic equations in reifenberg domains abstract in this paper we establish the global estimates for divergence form elliptic equations in the orlicz space l where is a bounded reifenberg domain in r n our result generalizes the w 1 p estimates

on the boundedness and stability of solutions to the affine variational inequality problem this paper investigates the boundedness and stability of solutions to the affine variational inequality problem the concept of a solution ray to a variational inequality defined by an affine mapping and on a closed convex set is introduced and characterized the connection of such a ray with the boundedness of the solution set of the given problem is explained in the case of the monotone affine variational inequality a complete description of the solution set is obtained which leads to a simplified characterization of the boundedness of this set as well as to a new error bound result for approximate solutions to such a variational problem the boundedness results are then combined with certain degree theoretic arguments to establish the stability of the solution set of an affine variational inequality problem

global gradient estimates for nonlinear equations of p laplace type from composite materials abstract we study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p growth nonlinearities from composite materials we assume that the domain is bounded and is composed of disjoint reifenberg flat domains and the nonlinearities have small bmo seminorms in each subdomain under these assumptions based on our new geometric observation for disjoint reifenberg domains and gradient estimates for nonlinear equations with measurable p growth nonlinearities we establish global w 1 q estimates for p q