lp gradient estimate for elliptic equations with high contrast conductivities in rn abstract uniform estimate for the solutions of elliptic equations with high contrast conductivities in r n is concerned the space domain consists of a periodic connected sub region and a periodic disconnected matrix block subset the elliptic equations have fast diffusion in the connected sub region and slow diffusion in the disconnected subset suppose 0 1 is the diameter of each matrix block and 2 0 1 is the conductivity ratio of the disconnected matrix block subset to the connected sub region it is proved that the w 1 p norm of the elliptic solutions in the connected sub region is bounded uniformly in when the l p norm of the elliptic solutions in the whole space is bounded uniformly in the w 1 p norm of the elliptic solutions in perforated domains is bounded uniformly in however the l p norm of the second order derivatives of the solutions in the connected sub region may not be bounded uniformly in

convergence in lp space for the homogenization problems of elliptic and parabolic equations in the plane we study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients first we propose homogenized expansions which are convolution forms of green function and given force term of elliptic equation then using local lp theory the growth rate of the perturbation of green function is found from the representation of elliptic solution by green function we estimate the convergence rate in lp space of the homogenized expansions to the exact solution finally we consider l2 0 t h1 or l 0 t convergence rate of the first order approximation for parabolic homogenization problems

a local error estimate of the method of multi scale asymptotic expansions for elliptic problems with rapidly oscillatory coefficients in this paper on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 we present a local error estimate of the method of multi scale asymptotic expansions for second order elliptic problems with rapidly oscillatory coefficients

heat kernel estimates and riesz transforms on some riemannian covering manifolds consider a riemannian manifold m which is a galois covering of a compact manifold with nilpotent deck transformation group g for the laplace operator on m we prove a precise estimate for the gradient of the heat kernel and show that the riesz transforms are bounded in l p m 1 p we also obtain estimates for discrete oscillations of the heat kernel and boundedness of discrete riesz transform operators which are defined using the action of g on m

green s matrices and boundary estimates in parabolic homogenization abstract for a family of second order parabolic systems in divergence form with rapidly oscillating periodic and time dependent coefficients we establish the pointwise estimates of the green s matrices we investigate uniform boundary lipschitz estimates and boundary holder estimates as well as non tangential maximum function estimates in c 1 or c 1 cylinders consequently we also obtain the agmon miranda maximum principle

elliptic and parabolic equations in fractured media abstract the elliptic and the parabolic equations with dirichlet boundary conditions in fractured media are considered the fractured media consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected subset to the connected sub region it is proved that the w 1 p norm of the elliptic and the parabolic solutions in the high permeability sub region are bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in for the elliptic and the parabolic equations in periodic perforated domains it is also shown that the w 1 p norm of their solutions are bounded uniformly in

random walks on discrete groups of polynomial volume growth let mu be a probability measure with finite support on a discrete group gamma of polynomial volume growth the main purpose of this paper is to study the asymptotic behavior of the convolution powers mu n of mu if mu is centered then we prove upper and lower gaussian estimates we prove a central limit theorem and we give a generalization of the berry esseen theorem these results also extend to noncentered probability measures we study the associated riesz transform operators the main tool is a parabolic harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov this inequality implies that the positive mu harmonic functions are constant finally we give a characterization of the mu harmonic functions which grow polynomially

a new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions abstract in this paper we consider second order elliptic problems with rapidly oscillating coefficients on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 wen ming he jun zhi cui a pointwise estimate on the 1 order approximation of g x 0 e ima j appl math 70 2005 241 269 we propose a new approximate method to solve these problems of course we present its error estimate

non uniform elliptic equations in convex lipschitz domains abstract non uniform elliptic equations in convex lipschitz domains are concerned the non smooth domains consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected matrix block subset to the connected sub region the w 1 p norm for p 1 of the elliptic solutions in the high permeability sub region is shown to be bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in roughly speaking if the sources in the low permeability subset are small enough the solutions in that subset are bounded uniformly in otherwise the solutions cannot be bounded uniformly in relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

uniform lipschitz estimates in bumpy half spaces this paper is devoted to the proof of uniform holder and lipschitz estimates close to oscillating boundaries for divergence form elliptic systems with periodically oscillating coefficients our main point is that no structure is assumed on the oscillations of the boundary in particular those oscillations are neither periodic nor quasiperiodic nor stationary ergodic we investigate the consequences of our estimates on the large scales of green and poisson kernels our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries which is an area of active research

directional homogenization of elliptic equations in non divergence form abstract the paper concerns regularities of solutions of elliptic equations in non divergence form with directional homogenization the main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization we establish uniform interior l p 1 p estimates for d 2 u e and uniform interior c 0 1 estimates for d d x u e where x are the non homogenization directions our results are optimal

quantitative estimates in reiterated homogenization abstract this paper investigates quantitative estimates in the homogenization of second order elliptic systems with periodic coefficients that oscillate on multiple separated scales we establish large scale interior and boundary lipschitz estimates down to the finest microscopic scale via iteration and rescaling arguments we also obtain a convergence rate in the l 2 space by the reiterated homogenization method

higher order linearization and regularity in nonlinear homogenization we prove large scale c regularity for solutions of nonlinear elliptic equations with random coefficients thereby obtaining a version of the statement of hilbert s 19th problem in the context of homogenization the analysis proceeds by iteratively improving three statements together i the regularity of the homogenized lagrangian l ii the commutation of higher order linearization and homogenization and iii large scale c 0 1 type regularity for higher order linearization errors we consequently obtain a quantitative estimate on the scaling of linearization errors a liouville type theorem describing the polynomially growing solutions of the system of higher order linearized equations and an explicit heterogenous analogue of the taylor series for an arbitrary solution of the nonlinear equations with the remainder term optimally controlled these results give a complete generalization to the nonlinear setting of the large scale regularity theory in homogenization for linear elliptic equations

a holder estimate for non uniform elliptic equations in a random medium abstract uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned the medium is separated by a random ensemble of simply closed interfaces into a connected sub region with high conductivity and a disconnected subset with low conductivity the elliptic equations whose diffusion coefficients depend on the conductivity have fast diffusion in the connected sub region and slow diffusion in the disconnected subset without a stationary ergodic assumption a uniform holder estimate in for the elliptic solutions is derived where is a realization of the random ensemble 0 1 is the length scale of the interfaces and 2 0 1 is the conductivity ratio of the disconnected subset to the connected sub region results show that if external sources are small enough in the disconnected subset the uniform holder estimate in holds in the whole domain if not it holds only in the connected sub region meanwhile the elliptic solutions change rapidly in the disconnected subset

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

on numerical approximation of a variational hemivariational inequality modeling contact problems for locking materials abstract this paper is devoted to numerical analysis of a new class of elliptic variational hemivariational inequalities in the study of a family of contact problems for elastic ideally locking materials the contact is described by the signorini unilateral contact condition and the friction is modeled by a nonmonotone multivalued subdifferential relation allowing slip dependence the problem involves a nonlinear elasticity operator the subdifferential of the indicator function of a convex set for the locking constraints and a nonconvex locally lipschitz friction potential solution existence and uniqueness result on the inequality can be found in migorski and ogorzaly 2017 in this paper we introduce and analyze a finite element method to solve the variational hemivariational inequality we derive a cea type inequality that serves as a starting point of error estimation numerical results are reported showing the performance of the numerical method

dirichlet to neumann and elliptic operators on c 1 kappa domains poisson and gaussian bounds abstract we prove poisson upper bounds for the heat kernel of the dirichlet to neumann operator with variable holder coefficients when the underlying domain is bounded and has a c 1 boundary for some 0 we also prove a number of other results such as gradient estimates for heat kernels and green functions g of elliptic operators with possibly complex valued coefficients we establish holder continuity of x y g up to the boundary these results are used to prove l p estimates for commutators of dirichlet to neumann operators on the boundary of c 1 domains such estimates are the keystone in our approach for the poisson bounds

convergence rates of neumann problems for stokes systems abstract in quantitative homogenization of the neumann problems for stokes systems with rapidly oscillating periodic coefficients this paper studies the convergence rates of the velocity in l 2 and h 1 as well as those of the pressure term in l 2 without any smoothness assumptions on the coefficients

uniform estimates in two periodic homogenization problems we analyze two partial differential equations that are posed on perforated domains we provide a priori estimates that do not depend on the size of the perforation a sequence of solutions is uniformly bounded in a sobolev space of regular functions the first homogenization problem concerns the laplace and the mean curvature operator with neumann boundary conditions we derive uniform lipschitz estimates for the solutions the result is used in the analysis of a free boundary system of fluid mechanics a contractive iteration yields the existence of solutions and uniform estimates the key is the use of function spaces that are different from the usual lp spaces 2000 john wiley sons inc

variational approach to homogenization of doubly nonlinear flow in a periodic structure this work deals with the homogenization of an initial and boundary value problem for the doubly nonlinear system d t w del z over right arrow g x t x epsilon 0 1 w is an element of alpha u x epsilon 0 2 z over right arrow is an element of gamma over right arrow del u x epsilon 0 3 here epsilon is a positive parameter alpha and gamma over right arrow are maximal monotone with respect to the first variable and periodic with respect to the second one the inclusions 0 2 and 0 3 are here formulated as null minimization principles via the theory of fitzpatrick mr 1009594 as epsilon 0 a two scale formulation is derived via nguetseng s notion of two scale convergence and a single scale homogenized problem is then retrieved c 2015 elsevier ltd all rights reserved

convergence rates in l 2 for elliptic homogenization problems we study rates of convergence of solutions in l2 and h1 2 for a family of elliptic systems mathcal l varepsilon with rapidly oscillating coefficients in lipschitz domains with dirichlet or neumann boundary conditions as a consequence we obtain convergence rates for dirichlet neumann and steklov eigenvalues of mathcal l varepsilon most of our results which rely on the recently established uniform estimates for the l2 dirichlet and neumann problems in kenig and shen math ann 350 867 917 2011 commun pure appl math 64 1 44 2011 are new even for smooth domains

uniform w 1 p estimates for systems of linear elasticity in a periodic medium abstract let l e be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients we obtain the uniform w 1 p estimate u e p c f p in a lipschitz domain in r n for solutions to the dirichlet problem l e u e div f in and u e 0 on where 1 p 1 2 1 2 n and c 0 are constants independent of e 0 the ranges are sharp for n 2 or 3

on second order periodic elliptic operators in divergence form we consider second order strongly elliptic operators with complex coecients in divergence form on r d we assume that the coecients are all periodic with a common period if the coecients are continu ous we derive gaussian bounds with the correct small and large time asymptotic behaviour on the heat kernel and all its holder deriva tives moreover we show that the first order riesz transforms are bounded on the lp spaces with p 2 h1 1i secondly if the coe cients are holder continuous we prove that the first order derivatives of the kernel satisfy good gaussian bounds then we establish that the second order derivatives exist and satisfy good bounds if and only if the coecients are divergence free or if and only if the second order riesz transforms are bounded finally if the third order derivatives exist with good bounds then the coecients must be constant

quantitative linearization results for the monge ampere equation this paper is about quantitative linearization results for the monge ampere equation with rough data we develop a large scale regularity theory and prove that if a measure is close to the lebesgue measure in wasserstein distance at all scales then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding poisson equation the main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures this is used in a campanato iteration which transfers the information through the scales

calderon zygmund estimates for stochastic homogenization abstract we prove quenched l p type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients

multiscale asymptotic method for steklov eigenvalue equations in composite media in this paper we consider the multiscale analysis of a steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure there are mainly two new results in the present paper first we obtain the convergence rate with varepsilon 1 2 for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the steklov eigenvalue problem second the boundary layer solution is defined numerical simulations are then carried out to validate the above theoretical results

homogenization in perforated domains and interior lipschitz estimates abstract we establish interior lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale e by establishing h 1 convergence rates for such solutions the interior estimates are derived directly without the use of compactness via an argument presented in 3 that was adapted for elliptic equations in 2 and 11 as a consequence we derive a liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains

a priori estimate for non uniform elliptic equations abstract a priori estimate for non uniform elliptic equations with periodic boundary conditions is concerned the domain considered consists of two sub regions a connected high permeability region and a disconnected matrix block region with low permeability let denote the size ratio of one matrix block to the whole domain it is shown that in the connected high permeability sub region the holder and the lipschitz estimates of the non uniform elliptic solutions are bounded uniformly in but holder gradient estimate and l p estimate of the second order derivatives of the solutions in general are not bounded uniformly in

boundary estimates in homogenization of systems of elasticity on lipschitz domains l p theory abstract the primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization in lipschitz domains as a consequence for d 4 we prove that the l p neumann and l p dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2 d 1 d 1 p 2 and 2 p 2 d 1 d 3 respectively

uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the neumann boundary problem abstract in this paper we mainly employed the idea of the previous paper 34 to study the sharp uniform w 1 p estimates with 1 p for more general elliptic systems with the neumann boundary condition on a bounded c 1 domain arising in homogenization theory based on the skills developed by z shen in 27 and by t suslina in 31 32 we also established the l 2 convergence rates on a bounded c 1 1 domain and a lipschitz domain respectively here we found a rough version of the first order correctors see 1 12 which can unify the proof in 27 and 32 it allows us to skip the corresponding convergence results on r d that are the preconditions in 31 32 our results can be regarded as an extension of 23 developed by c kenig f lin z shen as well as of 32 investigated by t suslina

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

applications of fourier analysis in homogenization of dirichlet problem i pointwise estimates in this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating dirichlet data our method is based on analysis of oscill

convergence rates and w1 p estimates in homogenization theory of stokes systems in lipschitz domains abstract concerned with the stokes systems with rapidly oscillating periodic coefficients we mainly extend the recent works in 19 20 to those in term of lipschitz domains the arguments employed here are quite different from theirs and the basic idea comes from 37 originally motivated by 23 27 33 we obtain an almost sharp o e ln r 0 e convergence rate in l 2 space and a sharp o e error estimate in l 2 d d 1 space by a little stronger assumption under the dimensional condition d 2 we also establish the optimal o e convergence rate on pressure terms in l 2 space then utilizing the convergence rates we can derive the w 1 p estimates uniformly down to microscopic scale e without any smoothness assumption on the coefficients where 1 p 1 2 1 2 d and is a positive constant independent of e combining the local estimates based upon vmo coefficients consequently leads to the uniform w 1 p estimates here the proofs do not rely on the well known compactness methods

the additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

homogenization and boundary layers this paper deals with the homogenization of elliptic systems with dirichlet boundary condition when the coefficients of both the system and the boundary data are e periodic we show that as e 0 the solutions converge in l2 with a power rate in e and identify the homogenized limit system due to a boundary layer phenomenon this homogenized system depends in a non trivial way on the boundary our analysis answers a longstanding open problem raised for instance in 7 it extends substantially previous results obtained for polygonal domains with sides of rational slopes as well as our previous paper 12 where the case of irrational slopes was considered

mesoscopic higher regularity and subadditivity in elliptic homogenization we introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form unlike previous works on the topic our method does not use concentration inequalities such as poincare or logarithmic sobolev inequalities in the probability space and relies instead on a higher ck k 1 regularity theory for solutions of the heterogeneous equation which is valid on length scales larger than a certain specified mesoscopic scale this regularity theory which is of independent interest allows us to in effect localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument the fluctuations of the energy are then tightly controlled using subadditivity the convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes that is it quantifies the weak convergence of these quantities which yields by a new multiscale poincare inequality quantitative estimates on the sublinearity of the corrector

homogenization of nonlinear equations with mixed boundary conditions in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings

linear elliptic equations in composite media with anisotropic fibres abstract linear elliptic equations in composite media with anisotropic fibres are concerned the media consist of a periodic set of anisotropic fibres with low conductivity included in a connected matrix with high conductivity inside the anisotropic fibres the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions the coefficients of the elliptic equations depend on the conductivity this work is to derive the holder and the gradient l p estimates uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix for the solutions of the elliptic equations furthermore it is shown that inside the fibres the solutions have higher regularity along the fibres than in the transverse directions

convergence rates and interior estimates in homogenization of higher order elliptic systems abstract this paper is concerned with the quantitative homogenization of 2m order elliptic systems with bounded measurable rapidly oscillating periodic coefficients we establish the sharp o e convergence rate in w m 1 p 0 with p 0 2 d d 1 in a bounded lipschitz domain in r d as well as the uniform large scale interior c m 1 1 estimate with additional smoothness assumptions the uniform interior c m 1 1 w m p and c m 1 estimates are also obtained as applications of the regularity estimates we establish asymptotic expansions for fundamental solutions

a multiscale algorithm for radiative heat transfer equation with rapidly oscillating coefficients this paper as a continued work of huang and cao 2014 discusses the multiscale computation of the radiative heat transfer in composite materials or porous media a novel multiscale asymptotic expansion is presented and an explicit rate of convergence is derived we develop a multiscale algorithm for solving this kind of problem a fully implicit scheme is carefully studied and an iterative algorithm is given the convergence of the iterative algorithm is proved by the fixed point method numerical results confirm the efficiency and accuracy of this approach and show that the novel multiscale asymptotic expansion is essential for the radiative dominated cases

optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form we prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form under strong independence assumptions on the coefficients we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential type bounds in probability like the theory of gloria and otto ann probab 39 3 779 856 2011 ann appl probab 22 1 1 28 2012 for divergence form equations the arguments rely on nonlinear concentration inequalities combined with certain estimates on the green s functions and derivative bounds on the correctors we obtain these analytic estimates by developing a c1 1 regularity theory down to microscopic scale which is of independent interest and is inspired by the c0 1 theory introduced in the divergence form case by the first author and smart ann sci ec norm super 4 49 2 423 481 2016

boundary korn inequality and neumann problems in homogenization of systems of elasticity this paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization we establish uniform optimal regularity estimates for solutions of neumann problems in a bounded lipschitz domain with l2 boundary data the proof relies on a boundary korn inequality for solutions of systems of linear elasticity and uses a large scale rellich estimate obtained in shen anal pde arxiv 1505 00694v2

bounded correctors in almost periodic homogenization we show that certain linear elliptic equations and systems in divergence form with almost periodic coefficients have bounded almost periodic correctors this is proved under a new condition we introduce which quantifies the almost periodic assumption and includes but is not restricted to the class of smooth quasiperiodic coefficient fields which satisfy a diophantine type condition previously considered by kozlov mat sb n s 107 149 199 217 1978 the proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by armstrong and shen pure appl math 2016 for equations with almost periodic coefficients this yields control on spatial averages of the gradient of the corrector which is converted into estimates on the size of the corrector itself via a multiscale poincare type inequality

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

convergence rate in homogenization of elliptic systems with singular perturbations we consider the optimal convergence rate in periodic homogenization of second order elliptic systems involving singular perturbations in bounded domains by introducing proper auxiliary functions we establish the sharp convergence rate in l2 by using the duality argument we consider the optimal convergence rate in periodic homogenization of second order elliptic systems involving singular perturbations in bounded domains by introducing proper auxiliary functions we establish the sharp convergence rate in l2 by using the duality argument

a characterization of truth functions in the nilpotent minimum logic abstract by introducing a new family of partitions into the n cube 0 1 n the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented so far only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left continuous but discontinuous t norm