hessian estimates in orlicz spaces for fourth order parabolic systems in non smooth domains abstract we establish the global hessian estimate in orlicz spaces for a fourth order parabolic system with discontinuous tensor coefficients in a non smooth domain under the assumptions that the coefficients have small weak bmo semi norms the boundary of a domain is reifenberg flat for 0 small and the given young function satisfies some moderate growth condition as a corollary we obtain an optimal global w 2 p regularity for such a system

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

quasilinear elliptic equations with bmo coefficients in lipschitz domains we obtain a global estimate for the weak solution to an elliptic partial differential equation of laplacian type with bmo coefficients in a lipschitz domain with small lipschitz constant

uniformly nondegenerate elliptic equations with partially bmo coefficients in nonsmooth domains abstract a global estimate in weighted lorentz sobolev spaces is obtained for the weak solutions to divergence form uniformly nondegenerate elliptic equations over a bounded nonsmooth domain here the leading coefficients are assumed to be merely measurable in one variable and have small bmo semi norms in the remaining variables under the assumption that one variable direction is perpendicular to the boundary points which is close to the boundary while a geometric assumption on the boundary is a locally bounded reifenberg flatness in addition we also investigate regularities in lorentz morrey morrey and holder spaces for elliptic equations under the same assumptions on the leading coefficients and the boundary of domain

nonlinear gradient estimates for parabolic problems with irregular obstacles abstract we establish the natural calderon zygmund theory for solutions to parabolic variational inequalities satisfying an irregular obstacle constraint and involving degenerate singular operators in divergence form of general type and proving that the spatial gradient of solutions is as integrable as both the spatial gradient of the obstacles and the inhomogeneous terms under the assumption that the involved nonlinearities have small a bmo semi norm in the spatial variables while they are allowed to be merely measurable in the time variable

weighted estimates for generalized steady stokes systems in nonsmooth domains we consider a generalized steady stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted lq space for 2 q we prove the global weighted lq estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small bounded mean oscillation semi norms and the domain is sufficiently flat in the reifenberg sense on the other hand a given weight is assumed to belong to a muckenhoupt class our result generalizes the global w1 q estimate of calderon zygmund with respect to the lebesgue measure for the stokes system in a lipschitz domain

local holder regularity for solutions of elliptic systems in this note we prove local l p regularity for the highest order derivatives of an elliptic system of arbitrary order in nondivergence form where the coefficients of the principal part are taken in the space of sarason vanishing mean oscillation vmo lower order coefficients and the known term belong to suitable lebesgue spaces as

elliptic equations with measurable coefficients in reifenberg domains abstract we prove w 1 p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in n 1 directions we assume the boundary to be reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary

w1 p estimates for elliptic problems with neumann boundary conditions in lipschitz domains abstract we study w 1 p estimates in lipschitz domains for second order elliptic equations and systems of divergence form with real valued bounded measurable coefficients for any fixed p 2 we prove that a weak reverse holder inequality implies the w 1 p estimates for solutions with neumann boundary conditions as a result we are able to show that if the coefficient matrix of elliptic equation is symmetric and in vmo r n the w 1 p estimate holds for 3 2 e p 3 e if n 3 and for 4 3 e p 4 e if n 2 and for elliptic systems the w 1 p estimate is true for 2 n n 1 e p 2 n n 1 e

fuzzy integers fuzzy rationals and other subspaces of the fuzzy real line abstract it is the purpose of this note to draw the attention to some noteworthy subspaces of the fuzzy real line m r which can be characterized by means of classical well known sequence spaces and function spaces

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

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

uniform regularity estimates in homogenization theory of elliptic system with lower order terms abstract in this paper we extend the uniform regularity estimates obtained by m avellaneda and f lin in 3 6 to the more general second order elliptic systems in divergence form l e e 0 with rapidly oscillating periodic coefficients we establish not only sharp w 1 p estimates holder estimates lipschitz estimates and non tangential maximal function estimates for the dirichlet problem on a bounded c 1 domain but also a sharp o e convergence rate in h 0 1 by virtue of the dirichlet correctors moreover we define the green s matrix associated with l e and obtain its decay estimates we remark that the well known compactness methods are not employed here instead we construct the transformations 1 11 to make full use of the results in 3 6

holder regularity of the solutions of some classes of elliptic systems in convex nonsmooth domains abstract we prove at first the holder continuity up to the boundary of a convex nonsmooth domain of the weak solution of a linear second order elliptic system in divergence form with discontinuous coefficients under a suitable condition on the dispersion of the eigenvalues of the coefficients matrix we apply the above result to some classes of quasilinear and nonlinear elliptic systems moreover we show existence and uniqueness of a very weak solution of a linear second order elliptic system with discontinuous coefficients and l 1 right hand side extending an analogous result for one single equation due to stampacchia

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

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

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

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

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

invariant elliptic estimates abstract in this paper we revisit elliptic estimates invariant under domain expansion we improve the invariant elliptic estimates in the previous paper y cho t ozawa y shim calc var pde 34 2009 321 339 via gradient estimate and discuss an application to the lame system

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

interior hw 1 p estimates for divergence degenerate elliptic systems in carnot groups abstract let x 1 x q be the basis of the space of horizontal vector fields on a homogeneous carnot group g r n q n we consider the following divergence degenerate elliptic system 1 n i j 1 q x i a i j x x j u i 1 q x i f i 1 2 n where the coefficients a i j are real valued bounded measurable functions defined in g satisfying the strong legendre condition and belonging to the space v m o l o c defined by the carnot caratheodory distance induced by the x i s we prove interior h w 1 p estimates 2 p for weak solutions to the system