new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

a study of boundedness in probabilistic normed spaces probabilistic normed spaces have been redefined by alsina schweizer and sklar we give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

on maximal regularity of type lp lq under minimal assumptions for elliptic non divergence operators we prove maximal regularity of type l p l q for operators in non divergence form with complex valued measurable coefficients on r n for a certain range of q which depends on dimension and the order of the operators this is done under the sole assumption that they generate an analytic semigroup in l q thus we give for this class of operators and this range of q a positive answer to brezis question whether generation of an analytic semigroup entails maximal l p regularity for other values of q we give several additional assumptions the proof relies on a result on maximal regularity under the assumption of suitable off diagonal bounds due to s blunck and the author which we improve here these off diagonal bounds are obtained by a modification of davies technique suited to cope with operators in non divergence form and they also imply new results on the scale of l q spaces in which a given operator generates an analytic semigroup we thus obtain a completely new approach to maximal l p regularity for operators of this kind in particular for those whose highest order coefficients belong to vmo moreover we obtain extensions of recent results due to kim and krylov for operators with measurable coefficients depending on one coordinate

use of abstract hardy spaces real interpolation and applications to bilinear operators this paper can be considered as the sequel of bernicot and zhao j func anal 255 1761 1796 2008 where the authors have proposed an abstract construction of hardy spaces h1 they shew an interpolation result for these hardy spaces with the lebesgue spaces here we describe a more precise result using the real interpolation theory and we clarify the use of hardy spaces then with the help of the bilinear interpolation theory we then give applications to study bilinear operators on lebesgue spaces these ideas permit us to study singular operators with singularities similar to those of bilinear calderon zygmund operators in a far more abstract framework as in the euclidean case

vertical versus conical square functions we study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator trans amer math soc 364 2012 no 10 5469 5489

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

countable products of probabilistic normed spaces abstract countable products of probabilistic normed spaces are introduced and studied in particular a comparison is made with the analogous constructions for probabilistic metric spaces

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

a maximal function characterization for hardy spaces associated to nonnegative self adjoint operators satisfying gaussian estimates abstract let l be a nonnegative self adjoint operator satisfying gaussian estimates on l 2 r n in this article we give an atomic decomposition for the hardy spaces h l max p r n in terms of the nontangential maximal functions associated with the heat semigroup of l and this leads eventually to characterizations of hardy spaces associated to l via atomic decomposition or the nontangential maximal functions the proof is based on a modification of a technique due to a calderon 6

numerical simulation of anisotropic surface diffusion with curvature dependent energy the aim of this paper is the numerical simulation of surface diffusion processes in the presence of a strong anisotropy and curvature dependence in the surface energy we derive semi implicit finite element discretizations based on a splitting into three second order equations the discretization we use yields indefinite linear systems for the nodal values of the height function the curvature concentration and the chemical potential we provide several numerical examples and parametric studies with respect to some of the parameters in the surface energy and with respect to the coverage the results to our knowledge the first that have been obtained for this model confirm theoretical predictions namely partial faceting of the surfaces with rounded corners

on maximal l p regularity abstract the aim of this paper is to propose weak assumptions to prove maximal l q regularity for cauchy problem d u d t t l u t f t mainly we only require off diagonal estimates on the real semigroup e t l t 0 to obtain maximal l q regularity the main idea is to use one kind of hardy space h 1 adapted to this problem and then use interpolation results these techniques permit us to prove weighted maximal regularity too

a new phase field model for strongly anisotropic systems we present a new phase field model for strongly anisotropic crystal and epitaxial growth using regularized anisotropic cahn hilliard type equations such problems arise during the growth and coarsening of thin films when the anisotropic surface energy is sufficiently strong sharp corners form and unregularized anisotropic cahn hilliard equations become ill posed our models contain a high order willmore regularization where the square of the mean curvature is added to the energy to remove the ill posedness the regularized equations are sixth order in space a key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation using the method of matched asymptotic expansions we show the convergence of our phase field model to the general sharp interface model we present two and three dimensional numerical results using an adaptive nonlinear multigrid finite difference method we find excellent agreement between the dynamics of the new phase field model and the sharp interface model the computed equilibrium shapes using the new model also match a recently developed analytical sharp interface theory that describes the rounding of the sharp corners by the willmore regularization

phase field theory of edges in an anisotropic crystal in the presence of sufficiently strong surface energy anisotropy the equilibrium shape of an isothermal crystal may include corners or edges models of edges have to date involved the regularization of the corresponding free boundary problem resulting in equilibrium shapes with smoothed out edges in this paper we take a new approach and consider how a phase field model which provides a diffuse description of an interface can be extended to the consideration of edges by an appropriate regularization of the underlying mathematical model using the method of matched asymptotic expansions we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy

surface evolution of elastically stressed films under deposition by a diffuse interface model we consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model the model is applicable to describe the self organization of nanostructures the influence of strain surface energies and kinetics on the surface evolution is considered a matched asymptotic analysis shows the formal convergence of an anisotropic viscous cahn hilliard model to a general surface evolution equation the system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models

abstract hardy spaces with variable exponents abstract in this paper we introduce the definition of abstract hardy spaces with variable exponents via the atomic decomposition and molecular decomposition we prove the continuity from our variable hardy space h a t o p into l p where p r n 0 1 0 p p 1 moreover we investigate the bilinear theory finally we give an application of the abstract hardy spaces with variable exponents

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

a regularized equation for anisotropic motion by curvature for realistic interfacial energies the equations of anisotropic motion by curvature exhibit backward parabolic behavior over portions of their domain thereby inducing phenomena such as the formation of facets and wrinkles in this paper a physically consistent regularized equation that may be used to analyze such phenomena is derived

interface evolution in three dimensions with curvature dependent energy and surface diffusion interface controlled evolution phase transitions epitaxial growth of elastic films when the interfacial energy is a nonconvex function of orientation the anisotropic curvature flow equation becomes backward parabolic to overcome the instability thus generated a regularization of the equation that governs the evolution of the interface is needed in this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk surface interactions the theory is based on a superficial mass balance configurational forces and couples consistent with superficial force and moment balances a mechanical version of the second law that includes via the configurational moments work that accompanies changes in the curvature of the interface a constitutive theory whose main ingredient is a positive definite isotropic quadratic dependence of the interfacial energy on the curvature tensor two special cases are investigated i the interface is a boundary between bulk phases or grains and ii the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface

weighted estimates for powers and smoothing estimates of schrodinger operators with inverse square potentials abstract let l a be a schrodinger operator with inverse square potential a x 2 on r d d 3 the main aim of this paper is to prove weighted estimates for fractional powers of l a the proof is based on weighted hardy inequalities and weighted inequalities for square functions associated to l a as an application we obtain smoothing estimates regarding the propagator e i t l a

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

finite element method for epitaxial growth with attachment detachment kinetics an adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth such problems consist of an adatom adsorbed atom diffusion equation on terraces of different height boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment and the normal velocity law for the motion of such boundaries determined by a two sided flux together with the one dimensional surface diffusion the problem is solved using two independent meshes a two dimensional mesh for the adatom diffusion and a one dimensional mesh for the boundary evolution the diffusion equation is discretized by the first order implicit scheme in time and the linear finite element method in space a technique of extension is used to avoid the complexity in the spatial discretization near boundaries all the elements are marked and the marking is updated in each time step to trace the terrace height the evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion its governing equation is solved by a semi implicit front tracking method using parametric finite elements simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem numerical tests on pure geometrical motion mass balance and the stability of a growing circular island demonstrate that the method is stable efficient and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period

equilibria and dislocations in epitaxial growth abstract an overview of recent analytical developments in epitaxial growth is presented the energy release via the onset of dislocations is addressed regularity of quasistatic equilibria is studied both in the absence and in the presence of dislocations morphological evolution of anisotropic epitaxially strained films is considered under the assumption that the surface evolves by surface diffusion under the influence of a chemical potential

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

second order convex maximum entropy approximants with applications to high order pde we present a new approach for second order maximum entropy max ent meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non uniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants we examine the performance of the proposed approximation scheme in the numerical solution by a direct galerkin method of a number of partial differential equations pdes including structural vibrations elliptic second order pdes and fourth order pdes for kirchhoff love thin shells and for a phase field model describing the mechanics of biomembranes the examples highlight the ability of the method to deal with non uniform node distributions and the high accuracy of the solutions surprisingly the first order meshfree max ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples even in the higher order pdes

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

self improving properties for abstract poincare type inequalities we study self improving properties in the scale of lebesgue spaces of generalized poincar e inequalities in the euclidean space we present an abstract setting where oscillations are given by certain operators e g approximations of the identity semigroups or mean value operators that have o diagonal decay in some range our results provide a unied theory that is applicable to the classical poincar e inequalities and furthermore it includes oscillations dened in terms of semigroups associated with second order elliptic operators as those in the kato conjecture in this latter situation we obtain a direct proof of the john nirenberg inequality for the associated bmo and lipschitz spaces of hmay hmm

surface diffusion of graphs variational formulation error analysis and simulation surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for graphs and derive a priori error estimates for a time continuous finite element discretization we also introduce a semi implicit time discretization and a schur complement approach to solve the resulting fully discrete linear systems after computational verification of the orders of convergence for polynomial degrees 1 and 2 we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time such as infinite slopes and cracks

motion of elastic thin films by anisotropic surface diffusion with curvature regularization in this paper we prove short time existence uniqueness and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two dimensional films this is achieved by using the h 1 gradient flow structure of the evolution law via de giorgi s minimizing movements this seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity

coarsening dynamics of the convective cahn hilliard equation abstract we characterize the coarsening dynamics associated with a convective cahn hilliard equation cch in one space dimension first we derive a sharp interface theory through a matched asymptotic analysis two types of phase boundaries kink and anti kink arise due to the presence of convection and their motions are governed to leading order by a nearest neighbors interaction coarsening dynamical system cds theoretical predictions on cds include the characteristic length l m for coarsening exhibits the temporal power law scaling t1 2 provided l m is appropriately small with respect to the peclet length scale l p binary coalescence of phase boundaries is impossible ternary coalescence only occurs through the kink ternary interaction two kinks meet an anti kink resulting in a kink direct numerical simulations performed on both cds and cch confirm each of these predictions a linear stability analysis of cds identifies a pinching mechanism as the dominant instability which in turn leads to kink ternaries we propose a self similar period doubling pinch ansatz as a model for the coarsening process from which an analytical coarsening law for the characteristic length scale l m emerges it predicts both the scaling constant c of the t1 2 regime i e l m ct 1 2 as well as the crossover to logarithmically slow coarsening as l m crosses l p our analytical coarsening law stands in good qualitative agreement with large scale numerical simulations that have been performed on cch