existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

divergence free wavelet bases on the hypercube free slip boundary conditions and applications for solving the instationary stokes equations we construct wavelet riesz bases for the usual sobolev spaces of divergence free functions on that have vanishing normals at the boundary we give a simultaneous space time variational formulation of the instationary stokes equations that defines a boundedly invertible mapping between a bochner space and the dual of another bochner space by equipping these bochner spaces by tensor products of temporal and divergence free spatial wavelets the stokes problem is rewritten as an equivalent well posed bi infinite matrix vector equation this equation can be solved with an adaptive wavelet method in linear complexity with best possible rate that under some mild besov smoothness conditions is nearly independent of the space dimension for proving one of the intermediate results we construct an eigenfunction basis of the stationary stokes operator

an elementary proof for a compact imbedding result in generalized electromagnetic theory propriete de plongement compact et certaines de ses consequences on etudie la validite du plongement suivant r 9 g e q 1 d q g l 2 q loc g

liouville theorems for the stokes equations with applications to large time estimates abstract we study liouville theorems for the non stationary stokes equations in exterior domains in r n under decay conditions for spatial variables as applications we prove that the stokes semigroup is a bounded analytic semigroup on l of angle 2 for n 3 we also prove large time estimates for n 2 with zero net force

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

maximal regularity of the stokes operator in an exterior domain with moving boundary and application to the navier stokes equations consider the navier stokes system on an exterior domain with moving boundary and dirichlet boundary conditions in 2003 saal proved that the stokes operator in a domain with moving boundary has the property of maximal regularity provided that the operator is invertible hence his result can be applied if the domain is bounded or by adding a shift to the stokes operator if the domain is unbounded or the time interval is finite in this paper we will generalize his result to a result global in time if the reference domain is an exterior domain finally we will apply this result to the navier stokes equations to obtain a global in time existence theorem for small data

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

convergence of euler stokes splitting of the navier stokes equations we consider approximation by partial time steps of a smooth solution of the navier stokes equations in a smooth domain in two or three space dimensions with no slip boundary condition for small k 0 we alternate the solution for time k of the inviscid euler equations with tangential boundary condition and the solution of the linear stokes equations for time k with the no slip condition imposed we show that this approximation remains bounded in h2 p and is accurate to order k in lp for p the principal difficulty is that the initial state for each stokes step has tangential velocity at the boundary generated during the euler step and thus does not satisfy the boundary condition for the stokes step the validity of such a fractional step method or splitting is an underlying principle for some computational methods 1994 john wiley sons inc

resolvent estimates in l p for the stokes operator in lipschitz domains we establish the l p resolvent estimates for the stokes operator in lipschitz domains in mathbb r d d geqq 3 for frac 1 p frac 1 2 frac 1 2d varepsilon the result implies that the stokes operator in a three dimensional lipschitz domain generates a bounded analytic semigroup in l p for 3 2 e p 3 e this gives an affirmative answer to a conjecture of m taylor progr nonlinear differential equations appl vol 42 pp 320 334

an h s p text curl varomega estimate for the maxwell system we derive an h 0 s p text curl varomega estimate for the solutions of the maxwell type equations modeled with anisotropic and w s infty varomega regular coefficients here we obtain the regularity of the solutions for the integrability and smoothness indices p s in a plane domain characterized by the apriori lower upper bounds of a and the apriori upper bound of its holder semi norm of order s the proof relies on a perturbation argument generalizing groger s l p type estimate known for the elliptic problems to the maxwell system

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

on non standard finite difference models of reaction diffusion equations reaction diffusion equations arise in many fields of science and engineering often their solutions enjoy a number of physical properties we design in a systematic way new non standard finite difference schemes which replicate three of these properties the first property is the stability instability of the fixed points of the associated space independent equation this property is preserved by non standard one and two stage theta methods presented in the general setting of stiff or non stiff systems of differential equations schemes which preserve the principle of conservation of energy for the corresponding stationary equation second property are constructed by nonlocal approximation of nonlinear reactions assembling of theta methods in the time variable with energy preserving schemes in the space variable yields non standard schemes which under suitable functional relation between step sizes display the boundedness and positivity of the solution third property a spectral method in the space variable coupled with a suitable non standard scheme in the time variable is also presented numerical experiments are provided

non standard finite differences schemes for generalized reaction diffusion equations reaction diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms in this work we design in a systematic way non standard finite differences fd schemes for a class of reaction diffusion equations of the form 1x sddx x sdudx f u x x 0 1 where s is the shape power that accounts for the complexity of the domain geometry the proposed fd scheme that is derived from a green s function formulation replicates the underlying geometry and reduces to traditional fd schemes for sufficiently small values of the grid spacing numerical results show that the non standard fd scheme offers smaller approximation errors with respect to traditional schemes specially for coarse grids

the stokes operator in weighted l q spaces ii weighted resolvent estimates and maximal l p regularity in this paper we establish a general weighted l q theory of the stokes operator mathcal a q omega in the whole space the half space and a bounded domain for general muckenhoupt weights omega in a q we show weighted l q estimates for the stokes resolvent system in bounded domains for general muckenhoupt weights these weighted resolvent estimates imply not only that the stokes operator mathcal a q omega generates a bounded analytic semigroup but even yield the maximal l p regularity of mathcal a q omega in the respective weighted l q spaces for arbitrary muckenhoupt weights omega in a q this conclusion is archived by combining a recent characterisation of maximal l p regularity by mathcal r bounded families due to weis operator valued fourier multiplier theorems and maximal l p regularity preprint 1999 with the fact that for l q spaces mathcal r boundedness is implied by weighted estimates

boundary characteristic point regularity for navier stokes equations blow up scaling and petrovskii type criterion a formal approach the three dimensional 3d navier stokes equations less thanbrgreater than less thanbrgreater thanu t u del u del p delta u divu 0 in q 0 0 1 less thanbrgreater than less thanbrgr

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

the brezis nirenberg problem for the curl curl operator abstract we look for solutions e r 3 of the problem e e e p 2 e in e 0 on on a bounded lipschitz domain r 3 where denotes the curl operator in r 3 the equation describes the propagation of the time harmonic electric field e x e i t in a nonlinear isotropic material with e 2 0 where and e stand for the permeability and the linear part of the permittivity of the material the nonlinear term e p 2 e with p 2 is responsible for the nonlinear polarisation of and the boundary conditions are those for surrounded by a perfect conductor the problem has a variational structure and we deal with the critical value p for instance in convex domains or in domains with c 1 1 boundary p 6 2 is the sobolev critical exponent and we get the quintic nonlinearity in the equation we show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on 0 we develop a new critical point theory which allows to solve the problem and which enables us to treat more general anisotropic media as well as other variational problems

besov and triebel lizorkin regularity for the hodge decomposition and applications to magnetic potentials abstract the hodge decomposition is a fundamental result in the theory of differential forms it has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics for example it is important in the analysis of navier stokes and maxwell s equations it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field the regularity properties of the hodge decomposition have been studied in several works in the last decades most prominently it has been proved that sobolev regularity holds true in this paper we prove regularity for the hodge decomposition in a much larger scale of function spaces besov and triebel lizorkin spaces on compact riemannian manifolds with boundary we prove regularity in such spaces for the hodge morrey decomposition and the hodge morrey friedrichs decomposition we furthermore state and prove a refinement for the hodge morrey friedrichs decomposition with regularity that allows expressing every differential form as the sum of the differential of a form plus the co differential of another form plus a harmonic field belonging to a finite dimensional space this holds true even if the original form is barely differentiable with sth order of differentiability for every real number s 0 we finally provide an application to the study of magnetic potentials associated to magnetic fields more precisely we prove the existence of a magnetic potential associated to a magnetic field the former being one time more differentiable than the latter in terms of besov and triebel lizorkin regularity this is an important motivation for us since we have the intention to apply our results to further investigations in relativistic inverse time dependent scattering problems in quantum mechanics where besov and triebel lizorkin regularity for magnetic potentials is relevant

qualitatively stable finite difference schemes for advection reaction equations a systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection reaction equations in the case of hyperbolic fixed points a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions non hyperbolic fixed points are described with the help of two new monotonic properties the construction of schemes which preserve these properties being done by nonlocal approximation of nonlinear terms in the reaction terms

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

on the initial boundary value problem for a bingham fluid in a three dimensional domain the initial boundary value problem associated with the motion of a bingham fluid is considered the existence and uniqueness of strong solution is proved under a certain assumption on the data it is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time periodic o introduction the purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a bingham fluid in a bounded three dimensional domain a bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached since the motion is governed by two entirely different stress strain relations depending on the state of stresses the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the 1 rlglcl zone the initial boundary value problem we shall study is formulated as 0 l it uj u a u w u b u u w j w j u f w a in 0 t for each test function w such that v w 0 in q and w 0 on aq 0 2 v u 0 in q x 0 t 0 3 u 0 on aq x 0 t 0 4 8 x 0 vo x in q here q is a bounded domain in r3 with smooth boundary aq a x t denotes the velocity of the fluid and f x t stands for external force we assume that the density the yield limit and the viscosity are positive constants in particular the received by the editors august 15 1986 1980 mathematics subject classijication 1985 revision primary 35q20 35b10 35b40 35b65 secondary 76a05 76d99 this research was supported by the air force office of scientific research under grant afosr86 0085 and by nsf grant dms 8521848 r 1987 american mathematical society 0002 9947 87 1 00 25 per page

perturbations of geodesic flows on surface of constant negative curvature and their mixing properties we consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature we find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones one condition says that the hamilton jacobi equation introduced in this work for the conjugation problem should admit a solution as a formal power series not necessarily convergent in the perturbation parameter the alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem the relation with the similar problems arising in the kam theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed as a byproduct of our analysis we obtain some results on the livscic guillemin kazhdan equation and on the fourier series for the sl 2 group we also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property with respect to the geodesic flow and to the invariant volume measure which is exponential with a universal exponent independent on the particular function equal to the curvature of the surface divided by 2 this result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow in this case we find that the decay rate is the inverse of the time up to logarithms

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

l q estimates for the stationary oseen equations on exterior domains we study the dirichlet problem for the stationary oseen equations on exterior smooth domains in rn n 2 our main results are the existence and uniqueness of weak and very weak solutions of the oseen equations satisfying appropriate lq estimates the uniqueness of very weak solutions is shown by utilizing recent results for very weak solutions of the stokes equations on bounded domains then for n 1 n 1 q the existence and lq estimates of very weak solutions in lq n is deduced by a duality argument from our existence result and d2 r estimates for strong solutions these results combined with some regularity results for very weak solutions enable us to prove the existence uniqueness and d1 r estimates of weak solutions in d1 r n l n 1 r n 1 r n where n 1 n r n 1

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

the regularity of the stokes operator and the fujita kato approach to the navier stokes initial value problem in lipschitz domains abstract we study the regularity of the navier stokes equations in arbitrary lipschitz domains

synchronisation of canonical measures for hyperbolic attractors under suitable conditions it is shown how to change the velocity of ac2 axioma attractor so that the sinai ruelle bowen measure coincides with the measure of maximal entropy these measures are obtained as limits of certain closed orbital measures

volume et entropie minimale des espaces localement symetriques m gromov introduit la notion de volume minimal d une variete riemannienne min vol m inf vol m g courbure sectionnelle de g 1 les questions naturelles sont alors i quelles varietes m verifient min vol m 0 min vol m 0 ii dans le second cas existe t il des metriques realisant le minimum lesquelles en guise de reponse a la seconde question on peut au mieux esperer que si m supporte une metrique naturelle elle realisera le volume minimal ceci conduit a la conjecture suivante posee par m gromov si m est une variete admettant une metrique hyperbolique le volume fini notee hyp alors min vol m vol m hyp le but de l article est de donner une reponse partielle a cette conjecture

stability results on interpolation scales of quasi banach spaces and applications we investigate the stability of fredholm properties on interpolation scales of quasi banach spaces this analysis is motivated by problems arising in pde s and several applications are presented

entropy production in thermostats ii we show that an arbitrary anosov gaussian thermostat close to equilibrium has positive entropy poduction unless the external field e has a global potential the configuration space is allowed to have any dimension and magnetic forces are also allowed we also show the following non perturbative result suppose a gaussian thermostat satisfies k w sigma frac14 e sigma 2 0 for every 2 plane where kw is the sectional curvature of the associated weyl connection and e sigma is the orthogonal projection of e onto then the entropy production of any srb measure is positive unless e has a global potential a related non perturbative result is also obtained for certain generalized thermostats on surfaces

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

on stokes operators with variable viscosity in bounded and unbounded domains we consider a generalization of the stokes resolvent equation where the constant viscosity is replaced by a general given positive function such a system arises in many situations as linearized system when the viscosity of an incompressible viscous fluid depends on some other quantities we prove that an associated stokes like operator generates an analytic semi group and admits a bounded h calculus which implies the maximal l q regularity of the corresponding parabolic evolution equation the analysis is done for a large class of unbounded domains with w 2 frac1r r boundary for some r d with r q q in particular the existence of an l q helmholtz projection is assumed

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

discretizations of nonlinear differential equations using explicit nonstandard methods abstract in a recent paper chen and solis investigated the appearance of spurious solutions when first order odes are discretized using runge kutta schemes they concluded that the reliability of the numerical solutions to a particular ode could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions we demonstrate that by using nonstandard schemes all the difficulties found by chen and solis can be eliminated and that qualitatively correct numerical solutions are obtained for all values of the step size we illustrate these issues by applying nonstandard finite difference techniques to the logistic sine cubic and monod equations

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

existence of strong solutions and decay of turbulent solutions of navier stokes flow with nonzero dirichlet boundary data abstract recently leray s problem of the l 2 decay of a special weak solution to the navier stokes equations with nonhomogeneous boundary values was studied by the authors exploiting properties of the approximate solutions converging to this solution in this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong energy inequality

exact controllability for the magnetohydrodynamic equations we study the local exact controllability of the steady state solutions of the magnetohydrodynamic equations the main result of the paper asserts that the steady state solutions of these equations are locally controllable if they are smooth enough we reduce the local exact controllability of the steady state solutions of the magnetohydrodynamic equations to the global exact controllability of the null solution of the linearized magnetohydrodynamic system via a x ed point argument the treatment of the reduced problem relies on two carleman type inequalities for the backward adjoint system c 2003 wiley periodicals inc

boundary layer methods for lipschitz domains in riemannian manifolds abstract we extend to the variable coefficient case boundary layer techniques that have been successful in the treatment of the laplace equation and certain other constant coefficient elliptic partial differential equations on lipschitz domains in euclidean space we treat the laplace operator on lipschitz domains in a manifold with c 1 metric tensor and study the dirichlet neumann and oblique derivative boundary problems

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

letter to the editor a green s function formulation of nonlocal finite difference schemes for reaction diffusion equations reaction diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms the aim of this work is to show that a green s function formulation of reaction diffusion pdes is a suitable framework to derive fd schemes incorporating both o h 2 accuracy and nonlocal approximations in the whole domain including boundary nodes by doing so the approach departs from a green s function formulation of the boundary value problem to pose an approximation problem based on a domain decomposition within each subdomain the corresponding integral equation is forced to have zero residual at given grid points different fd schemes are obtained depending on the numerical scheme used for computing the green s integral over each subdomain dirichlet and neumann boundary conditions are considered showing that the fd scheme based on the green s function formulation incorporates in a natural way the effects of boundary nodes in the discretization approximation