hilbert schemes and noncommutative deformations of type a kleinian singularities abstract let h k be a symplectic reflection algebra corresponding to a cyclic subgroup sl 2 c of order n and u k e h k e the spherical subalgebra of h k we show that for suitable k there is a filtered z n 1 algebra r such that 1 there is an equivalence of categories r qgr u k mod 2 there is an equivalence of categories gr r qgr coh hilb c 2 here coh hilb c 2 is the category of coherent sheaves on the hilbert scheme and for a graded algebra r we write r qgr for the quotient category of finitely generated graded r modules modulo torsion

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

optimization of the porous material described by the biot model abstract the paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids at the macroscopic level the porous material is described by the biot model defined in terms of the effective medium coefficients involving the drained skeleton elasticity the biot stress coupling the biot compressibility coefficients and by the hydraulic permeability of the darcy flow model by virtue of the homogenization these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid for the purpose of optimization the sensitivity analysis on the continuous level of the problem is derived we provide sensitivities of objective functions constituted by the biot model coefficients with respect to the underlying pore shape described by a b spline box which embeds the whole representative cell we consider material design problems in the framework of which the layout of a single representative cell is optimized then we propose a sequential linearization approach to the two scale problem in which local microstructures are optimized with respect to macroscopic design criteria numerical experiments are reported which include stiffness maximization with constraints guaranteeing minimum required permeability and vice versa issues of the design and anisotropy and the spline box parametrization are discussed in order to avoid remeshing a geometric regularization technique based on injectivity constraints is applied

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

calderon zygmund estimates for general elliptic operators with double phase abstract we derive calderon zygmund type estimates for the gradient of bounded distributional solutions to the problem div b x u d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where 1 p q and 0 a c the coefficient b x z is supposed to be small bmo with respect to x and merely continuous in z

homogenization based two scale modelling of ionic transport in fluid saturated deformable porous media abstract the paper deals with the homogenization and numerical modelling of deformable porous media saturated by two component electrolytes the model relevant to the microscopic scale describes steady states of the medium while reflecting essential physical phenomena namely electrochemical interactions in a dilute newtonian solvent under assumptions of a small external electrostatic field and slow flow the homogenization is applied to a linearized micromodel whereby the thermodynamic equilibrium represents the reference state due to the dimensional analysis scaling of the viscosity and electric permittivityis introduced so that the limit model retains the characteristic length associated with the pore size and the electric double layer thickness the homogenized model consists of two weakly coupled parts the steady flow of the electrolyte can be described in terms of a global pressure and streaming potentials of the two ions independently of the solid phase deformations which are computed afterwards for the fluid stress acting on pore walls the two scale model has been implemented in the sfepy finite element software the numerical results show dependence of the homogenized coefficients on the microstructure porosity by virtue of the corrector result of the homogenization microscopic responses in a local representative cell can be reconstructed from the macroscopic solutions

endpoint sobolev and bv continuity for maximal operators abstract in this paper we investigate some questions related to the continuity of maximal operators in w 1 1 and bv spaces complementing some well known boundedness results letting m be the one dimensional uncentered hardy littlewood maximal operator we prove that the map f m f is continuous from w 1 1 r to l 1 r in the discrete setting we prove that m b v z b v z is also continuous for the one dimensional fractional hardy littlewood maximal operator we prove by means of counterexamples that the corresponding continuity statements do not hold both in the continuous and discrete settings and for the centered and uncentered versions

regularity results for vectorial minimizers of a class of degenerate convex integrals abstract we establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with p q growth conditions we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity the results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

existence of renormalized solutions to elliptic equation in musielak orlicz space abstract we prove existence of renormalized solutions to general nonlinear elliptic equation in musielak orlicz space avoiding growth restrictions namely we consider div a x u f l 1 on a lipschitz bounded domain in r n the growth of the monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose is log holder continuity of m which results in good approximation properties of the space the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments

twisted multi homogeneous coordinate rings abstract artin and van den bergh have constructed noncommutative analogues of the homogeneous coordinate ring of a projective scheme we generalize their results to the multiprojective setting such multi homogeneous coordinate rings include tensor products of twisted homogeneous coordinate rings as well as their rees algebras finally we show that these last two examples are noetherian assuming ampleness conditions though the coordinate ring in general is not

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

the biot darcy brinkman model of flow in deformable double porous media homogenization and numerical modelling abstract in this paper we present the two level homogenization of the flow in a deformable double porous structure described at two characteristic scales the higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid the macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two level asymptotic analysis whereby a scaling ansatz is adopted to respect the pore size differences the first level upscaling of the fluid structure interaction problem yields a biot continuum describing the mesoscopic matrix coupled with the stokes flow in the channels the second step of the homogenization leads to a macroscopic model involving three equations for displacements the mesoscopic flow velocity and the micropore pressure due to interactions between the two porosities the macroscopic flow is governed by a darcy brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two phase medium expressions of the effective macroscopic parameters of the homogenized double porosity continuum are derived depending on the characteristic responses of the mesoscopic structure some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed the model has been implemented in the finite element code sfepy which is well suited for computational homogenization a numerical example of solving a nonstationary problem using mixed finite element method is included

holder continuity for nonlinear elliptic problem in musielak orlicz sobolev space abstract under appropriate assumptions on the n function the de giorgi process is presented by the tools recently developed in musielak orlicz sobolev space to prove the holder continuity of fully nonlinear elliptic problems as the applications the holder continuity of the minimizers for a class of the energy functionals in musielak orlicz sobolev spaces is proved and furthermore the local holder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided

regularity for double phase problems under additional integrability assumptions abstract we study the regularity of quasi minimizers or minimizers of functionals of double phase type in recent years baroni colombo and mingione have investigated regularity theory for bounded minimizers in baroni et al 2015 2018 colombo and mingione 2015 in this paper we assume the minimizers only belong to l space

regularity for minimizers of double phase functionals with mild transition and regular coefficients abstract we prove sharp regularity results for minimizers of the functional p w b x w d w p a x d w p log e d w d x with w w 1 1 p 1 a l a 0 and 0 b l p is a double phase functional with mild transition between d u p and d u p log e d u first under suitable conditions on the moduli of continuity of a and b we prove that local minimizers are of class c 0 for every 0 1 then that they are of class c 1 for some 0 provided the functions a and b are holder continuous

existence and multiplicity results for double phase problem abstract by variational method we obtain various existence and multiplicity results for the following double phase problem div u p 2 u a x u q 2 u f x u in u 0 on in particular we find a sign changing ground state solution some properties of double phase operator are also obtained

variable exponent functionals in image restoration abstract we study a functional with variable exponent 1 p x 2 which provides a model for image denoising and restoration here p x is defined by the gradient information in the observed image the diffusion derived from the proposed model is between total variation based regularization and gaussian smoothing the diffusion speed of the corresponding heat equation is tuned by the variable exponent p x the minimization problem and its associated flow in a weakened formulation are discussed the existence uniqueness stability and long time behavior of the proposed model are established in the variable exponent functional space w 1 p x experimental results illustrate the effectiveness of the model in image restoration

partial regularity for general systems of double phase type with continuous coefficients abstract we consider elliptic systems of double phase problems which satisfy a general structure with continuous coefficients and prove that weak solutions of such systems are holder continuous for any holder exponent 0 1 except for a measure zero set

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

multiple solutions for a class of double phase problem without the ambrosetti rabinowitz conditions abstract in the present paper in view of the variational approach we consider the existence and multiplicity of weak solutions for a class of the double phase problem div u p 2 u a x u q 2 u f x u in u 0 on where n 2 and 1 p q n firstly by the fountain and dual theorem with cerami condition we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f secondly we prove that this problem has at least one nontrivial solution for any parameter 0 small enough and also that the solution blows up in the sobolev norm as 0 finally by imposing additional assumptions on f we establish the existence of infinitely many solutions by using krasnoselskii s genus theory for the above equation

ample filters of invertible sheaves abstract let x be a scheme proper over a commutative noetherian ring a we introduce the concept of an ample filter of invertible sheaves on x and generalize the most important equivalent criteria for ampleness of an invertible sheaf we also prove the theorem of the base for x and generalize serre s vanishing theorem we then generalize results for twisted homogeneous coordinate rings which were previously known only when x was projective over an algebraically closed field specifically we show that the concepts of left and right ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian

noncommutative ampleness for multiple divisors abstract the twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of artin van den bergh and others chan generalized this construction to the multi homogeneous case using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme from this he derives that certain multi homogeneous rings such as tensor products of twisted homogeneous coordinate rings are right noetherian we show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness thus we find under natural hypotheses that multi homogeneous coordinate rings are noetherian and have integer gk dimension

modeling nonlinear phenomena in deforming fluid saturated porous media using homogenization and sensitivity analysis concepts a new computationally efficient nonlinear model of fluid saturated porous media two scale modelling of the biot darcy continuum based on the homogenization deformation dependent coefficients are computed using the sensitivity analysis nonlinear effects captured without any need to update local microstructures nonlinearity is pronounced especially for complex geometries of pores homogenization of heterogeneous media with nonlinear effects leads to computationally complex problems requiring updating local microstructures and solving local auxiliary problems to compute characteristic responses in this paper we suggest how to circumvent such a computationally expensive updating procedure while using an efficient approximation scheme for the local homogenized coefficients so that the complexity of the whole two scale modeling is reduced substantially we consider the deforming porous fluid saturated media described by the biot model the proposed modeling approaches are based on the homogenization of the quasistatic fluid structure interaction whereby differentiation with respect to the microstructure deformation is used as a tool for linearization assuming the linear kinematics framework the physical nonlinearity in the biot continuum is introduced in terms of the deformation dependent material coefficients which are approximated as linear functions of the macroscopic response these functions are obtained by the sensitivity analysis of the homogenized coefficients computed for a given geometry of the porous structure which transforms due to the local deformation the deformation dependent material coefficients approximated in this way do not require any solving of local microscopic problems for updated configurations it appears that difference between the linear and nonlinear models depends significantly on the specific microstructure of the porous medium this observation is supported by numerical examples

ample filters and frobenius amplitude abstract let x be a projective scheme over a field we show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the frobenius morphism we also compare various definitions of ample locally free sheaf and show that the vanishing given by the frobenius morphism is in a certain sense the strongest possible our work can be viewed as various generalizations of the serre vanishing theorem

noncommutative cyclic covers and maximal orders on surfaces abstract in this paper we construct various examples of maximal orders on surfaces including some del pezzo orders some ruled orders and some numerically calabi yau orders the method of construction is a noncommutative version of the cyclic covering trick these noncommutative cyclic covers are very computable and we give a formula for their ramification data this often allows us to determine if a maximal order described via ramification data can be constructed as a noncommutative cyclic cover the construction also has applications to brauer severi varieties and in the quaternion case we show how to obtain some brauer severi varieties from g hilbert schemes of p 1 bundles

regularity for multi phase variational problems abstract we prove c 1 regularity for local minimizers of the multi phase energy w d w p a x d w q b x d w s d x under sharp assumptions relating the couples p q and p s to the holder exponents of the modulating coefficients a and b respectively

double phase image restoration abstract in this paper we explore the potential of the double phase functional in an image processing context to this end we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained by convergence or relaxation of regularized functionals a central tool is a capped fractional maximal function of the derivative of bv functions