the mixed problem in lipschitz domains with general decompositions of the boundary this paper continues the study of the mixed problem for the laplacian we consider a bounded lipschitz domain rn n 2 with boundary that is decomposed as d n d and n disjoint we let denote the boundary of d relative to and impose conditions on the dimension and shape of and the sets n and d under these geometric criteria we show that there exists p0 1 depending on the domain such that for p in the interval 1 p0 the mixed problem with neumann data in the space lp n and dirichlet data in the sobolev space w 1 p d has a unique solution with the non tangential maximal function of the gradient of the solution in lp we also obtain results for p 1 when the dirichlet and neumann data comes from hardy spaces and a result when the boundary data comes from weighted sobolev spaces

the l p regularity problem on lipschitz domains this paper contains two results on the l p regularity problem on lipschitz domains for second order elliptic systems and 1 2 the solvability of the l p regularity problem is equivalent to a weak reverse holder condition with exponent p

finite element error estimates for 3d exterior incompressible flow with nonzero velocity at infinity we consider a stationary incompressible navier stokes flow in a 3d exterior domain with nonzero velocity at infinity in order to approximate this flow we use the stabilized p1 p1 finite element method proposed by rebollo numer math 79 283 319 1998 following an approach by guirguis and gunzburger model math anal numer 21 445 464 1987 we apply this method to the navier stokes system with oseen term in a truncated exterior domain under a pointwise boundary condition on the artificial boundary this leads to a discrete problem whose solution approximates the exterior flow as is shown by error estimates

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

enumerations of the hamiltonian walks on a cubic sublattice a massively parallel supercomputer was used to exhaustively enumerate all of the hamiltonian walks for simple cubic sublattices of four different sizes up to 3 4 4 the behaviour of the logarithm of the number of walks was found to be linear in the number of vertices in the lattice the linear fit is shown to agree also with the asymptotic limit of the flory mean field theoretical estimate thus we suggest that the fit obtained yields the number of walks for any size fragment of the cubic lattice to logarithmic accuracy the significance of this result to the validity of polymer models is also discussed

existence and uniqueness for a semilinear elliptic problem on lipschitz domains in riemannian manifolds ii abstract for a general class of semilinear equations in a lipschitz domain where m is a smooth compact riemannian manifold we establish the existence and uniqueness of solutions to dirichlet and neumann boundary problems the main contribution of this paper is that we consider rough boundary data that are typically just functions results of this type for the linear equation are typically obtained using singular integral techniques

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

solutions of the divergence operator on john domains if rn is a bounded domain the existence of solutions u w01 p of divu f for f lp with vanishing mean value and 1 p is a basic result in the analysis of the stokes equations it is known that the result holds when is a lipschitz domain and that it is not valid for domains with external cusps in this paper we prove that the result holds for john domains our proof is constructive the solution u is given by an explicit integral operator acting on f to prove that u w01 p we make use of the calderon zygmund singular integral operator theory and the hardy littlewood maximal function for domains satisfying the separation property introduced in s buckley p koskela sobolev poincare implies john math res lett 2 5 1995 577 593 and 1 p n we also prove a converse result thus characterizing in this case the domains for which a continuous right inverse of the divergence exists in particular our result applies to simply connected planar domains because they satisfy the separation property

commutator estimates for the dirichlet to neumann map of stokes systems in lipschitz domains abstract in the paper we establish commutator estimates for the dirichlet to neumann map of stokes systems in lipschitz domains the approach is based on dahlberg s bilinear estimates and the results may be regarded as an extension of 8 21 to stokes systems

green s matrices of second order elliptic systems with measurable coefficients in two dimensional domains we study green s matrices for divergence form second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains we establish existence uniqueness and pointwise estimates of green s matrices

weighted lq estimates for stationary stokes system with partially bmo coefficients abstract we prove the unique solvability of solutions in sobolev spaces to the stationary stokes system on a bounded reifenberg flat domain when the coefficients are partially bmo functions i e locally they are merely measurable in one direction and have small mean oscillations in the other directions using this result we establish the unique solvability in muckenhoupt type weighted sobolev spaces for the system with partially bmo coefficients on a reifenberg flat domain we also present weighted a priori l q estimates for the system when the domain is the whole euclidean space or a half space

gradient estimates for stokes systems with dini mean oscillation coefficients abstract we study the stationary stokes system in divergence form the coefficients are assumed to be merely measurable in one direction and have dini mean oscillations in the other directions we prove that if u p is a weak solution of the system then d u p is bounded and its certain linear combinations are continuous we also prove a weak type 1 1 estimate for d u p under a stronger assumption on the l 1 mean oscillation of the coefficients the corresponding results up to the boundary on a half ball are also established these results are new even for elliptic equations and systems

navier stokes equations on lipschitz domains in riemannian manifolds the navier stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous incompressible fluid one ingredient in the analysis of this system is the stationary linear system known as the stokes system a boundary value problem bvp that will be described in detail in the next section layer potential methods in smoothly bounded domains in euclidean space have proven useful in the analysis of the stokes system starting with work of odqvist and lichtenstein and including work of solonnikov and many others see the discussion in chapter iii of 10 and in 17 for the case of flow in regions with smooth boundary a treatment based on the modern language of pseudodifferential operators can be found in 18 in 1988 e fabes c kenig and g verchota 6 extended this classical layer potential approach to cover lipschitz domains in euclidean space in 6 the main result concerning the constant coefficient stokes system on lipschitz domains with connected boundary in euclidean space is the treatment of the l dirichlet boundary value problem and its regular version to achieve this the authors solve certain auxiliary neumann type problems and then exploit the duality between these and the original bvp s at the level of boundary integral operators p deuring and w von wahl 4 made use of the analysis in 6 to demonstrate the short time existence of solutions to the navier stokes equations in bounded lipschitz domains in threedimensional euclidean space it was necessary in 4 to include the hypothesis that the boundary be connected the hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on lipschitz domains it was certainly natural to speculate that this restriction was an artifact of the methods used and not partly supported by nsf grant dms 9870018 partially supported by nsf grant dms 9877077 1991mathematics subject classification primary 35q30 76d05 35j25 secondary 42b20 45e05

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

finite dimensional regulators for a class of regular hyperbolic pde systems abstractin this paper the output regulation problem is addressed for a class of linear hyperbolic infinite dimensional systems with spatially varying coefficients modelling a large class of convection dominated transport reaction systems in particular distributed parameter systems with bounded input and unbounded output operators are considered first we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation based on the assumption that the hyperbolic system satisfies the exponential stability conditions the main manuscript contribution is the development of two novel finite dimensional regulators output and error feedback regulators such that the controlled output of the plant tracks a reference signal generated by a known signal process exosystem in order to guarantee the feasibility of the proposed regulators the solvability of the corresponding sylvester equations is discusse

on enumeration of hilbert like curves we present an analytical method to explicitly enumerate all self similar space filling curves similar to hilbert curve and find their number grows with length l as this presents a first step in the exact characterization of the crumpled globule ensemble relevant for dense topologically constrained polymer matter and dna folding moreover this result gives a stringent lower bound on the number of hamiltonian walks on a simple cubic lattice additionally we compute the exact number of crumpled curves with arbitrary endpoints and the closed crumpled curves on a cube

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

exact enumeration of hamiltonian circuits walks and chains in two and three dimensions we present an algorithm for enumerating exactly the number of hamiltonian chains on regular lattices in low dimensions by definition these are sets of k disjoint paths whose union visits each lattice vertex exactly once the well known hamiltonian circuits and walks appear as the special cases k 0 and k 1 respectively in two dimensions we enumerate chains on l l square lattices up to l 12 walks up to l 17 and circuits up to l 20 some results for three dimensions are also given using our data we extract several quantities of physical interest

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

boundary integral approach for the mixed dirichlet robin boundary value problem for the nonlinear darcy forchheimer brinkman system abstract the purpose of this paper is to provide the solvability and uniqueness result to the mixed dirichlet robin boundary value problem for the nonlinear darcy forchheimer brinkman system in a bounded two dimensional lipschitz domain first we obtain a well posedness result for the linear brinkman system with dirichlet neumann boundary conditions by reducing the problem to the system of boundary integral equations based on the fundamental solution of the brinkman system and by analyzing this system employing a variational approach the result is extended afterwards to the poisson problem for the brinkman system and to dirichlet robin boundary conditions using the newtonian potential and the linearity of the solution operator further we study the nonlinear darcy forchheimer brinkman boundary value problem of dirichlet and robin type

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

variable lorentz estimate for conormal derivative problems of stationary stokes system with partially bmo coefficients abstract we prove a global calderon zygmund type estimate in the framework of lorentz spaces for the variable power of the gradients to the solution pair u p of the conormal derivative problems of stationary stokes system it is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables the boundary of domain belongs to the reifenberg flatness and the variable exponents p x satisfy the log holder continuity

layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains the symbolic calculus of pseudodifferential operators of mellin type is applied to study layer potentials on a plane domain q whose boundary 9q is a curvilinear polygon a singularity type is a zero of the determinant of the matrix of symbols of the mellin operators and can be used to calculate the bad values of p for which the system is not fredholm on lp 9q using the method of layer potentials we study the singularity types of the system of elastostatics lu uau au vdivu 0 in a plane domain q whose boundary 9k2 is a curvilinear polygon here u 0 and u a oo when a oo the system is the stokes system of hydrostatics for the traction double layer potential we show that all singularity types in the strip 0 re z i lie in the interval 1 1 so that the system of integral equations is a fredholm operator of index 0 on lp aq for all p 2 p oo the explicit dependence of the singularity types on a and the interior angles 0 of aq is calculated the singularity type of each corner is independent of a if the corner is nonconvex introduction recently there has been considerable interest in using layer potentials to solve lp boundary value problems for elliptic operators and systems on a lipschitz domain q in r for the systems of elastostatics dkv and hydrostatics fkv dahlberg fabes kenig and verchota have used rellich type identities to prove that the double layer potential integral equations yield a fredholm operator of index 0 on l 2 q for p 2 only limited information is available on the boundary integral equations for general lipschitz domains in rn the general problem of the notion of symbol on the boundary of a general lipschitz domain is still very much open in this paper we treat a very special case a curvilinear polygonal domain in r in this 2 dimensional case a precise symbolic calculus of pseudodifferential operators of mellin type is available we show that certain double layer boundary integral equations yield operators which for all p 2 p 00 are received by the editors december 14 1988 1980 mathematics subject classification 1985 revision primary 35j25 45e05 research partially supported by the italian cnr 1990 american mathematical society 0002 9947 90 1 00 25 per page

finite dimensional compensator design for parabolic distributed systems with point sensors and boundary input a compensator design is proposed for a large class of parabolic distributed systems basically it is a modal approach utilizing known finite dimensional algorithms for observer and stabilizer design the observer is constructed from observations taken from point or averaging sensors and the compensator is a feedback of this observer and can be a distributed or boundary implementation explicit sufficient conditions are given for the convergence of this scheme

on the navier problem for the stationary navier stokes equations the navier problem is to find a solution of the steady state navier stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary if is exterior it is required that the velocity converges to an assigned constant vector u0 at infinity we prove that a solution exists in a bounded domain provided a l2 is less than a computable positive constant and is unique if a w1 2 2 s l2 is suitably small as far as exterior domains are concerned we show that a solution exists if a l2 a u0 n l2 is small

regulator problem for infinite dimensional systems abstract in this paper a regulator problem is investigated for an infinite dimensional system with constant disturbances the regular problem is to determine a feedback control law which stabilizes and regulates the system a design procedure of a regulator is proposed which can be realized in finite dimensional theories and techniques for an infinite dimensional system

a pi controller for distributed delay systems abstract a finite dimensional control theory is developed for the design of a proportional plus integral control law for tracking step command inputs in general autonomous time lag systems with distributed heredity in state control and output variables with respect to an ordinary differential equation describing the delay system unstable modes an auxiliary tracking problem is posed this auxiliary problem derives its importance from the fact that its solution directly yields a solution to the original tracking problem this point of contact with a finite dimensional system permits the application of well tested ordinary systems tracking theory to the tracking problem in time lag systems

a boundary multiplier fictitious domain method for the steady incompressible navier stokes equations we analyze the error of a fictitious domain method with boundary lagrange multiplier it is applied to solve a non homogeneous steady incompressible navier stokes problem in a domain with a multiply connected boundary the interior mesh in the fictitious domain and the boundary mesh are independent up to a mesh length ratio

fundamental solutions for stationary stokes systems with measurable coefficients abstract we establish the existence and the pointwise bound of the fundamental solution for the stationary stokes system with measurable coefficients in the whole space r d d 3 under the assumption that weak solutions of the system are locally holder continuous we also discuss the existence and the pointwise bound of the green function for the stokes system with measurable coefficients on where is an unbounded domain such that the divergence equation is solvable such a domain includes for example half space and an exterior domain

estimates for green functions of stokes systems in two dimensional domains abstract we prove the existence and pointwise bounds of the green functions for stationary stokes systems with measurable coefficients in two dimensional domains we also establish pointwise bounds of the derivatives of the green functions under a regularity assumption on the l 1 mean oscillations of the coefficients

output regulation problem for a class of regular hyperbolic systems this paper investigates the output regulation problem for a class of regular first order hyperbolic partial differential equation pde systems a state feedback and an error feedback regulator are considered to force the output of the hyperbolic pde plant to track a periodic reference trajectory generated by a neutrally stable exosystem a new explanation is given to extend the results in the literature to solve the regulation problem associated with the first order hyperbolic pde systems moreover in order to provide the closed loop stability condition for the solvability of the regulator problems the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper this paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator riccati equation

output and error feedback regulator designs for linear infinite dimensional systems abstract this manuscript addresses the output regulation problem for linear distributed parameter systems dpss with bounded input and unbounded output operators we introduce novel methods for the design of the output feedback and error feedback regulators in the output feedback regulator design the measurements available for the regulator do not belong to the set of controlled outputs the proposed output feedback regulator with the injection of the measurement y m t and reference y r t can realize both the plant and the exosystem states estimation disturbance rejection and reference signal tracking simultaneously moreover new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design the regulator parameters are easily configured to solve the output regulation problems and to ensure the stability of the closed loop systems the results are demonstrated via computer simulation in two types of representative systems the parabolic partial differential equation pde system and the first order hyperbolic pde system

a finite element boundary element method for three dimensional large displacement fluid structure interaction abstract this paper presents a hybrid finite element boundary element method for fluid structure interaction simulations of inflatable structures the flow model consists of the steady stokes equation which admits a boundary integral formulation the structure is represented by a kirchhoff love shell the boundary element approximation of the stokes equation reduces the flow problem to an integral equation on the actual structure configuration thus obviating the need for volumetric meshing of the strongly deforming fluid domain the stokes model moreover exhibits a lubrication effect that acts as an intrinsic mechanism to treat the ubiquitous self contact that occurs in inflation problems the aggregated fluid structure interaction problem composed of the boundary integral equation and the kirchhoff love shell connected by dynamic and kinematic interface conditions is approximated by means of isogeometric discretizations to accommodate the smoothness requirements on the approximation spaces imposed by the flexural rigidity in the kirchhoff love shell and to provide an accurate and smooth representation of the boundary for the boundary element method auxiliary results presented in this paper are 1 a parametrization free kirchhoff love formulation 2 establishment of a cubic relationship between distance and tractions due to the lubrication effect and 3 the interpretation of the lagrange multiplier pertaining to fluid incompressibility as the total excess pressure

exact enumeration of hamiltonian walks on the 4 4 4 cube and applications to protein folding hamiltonian walks on lattices are model systems for compact polymers such as proteins here we enumerate exactly the number of hamiltonian walks on the 4 4 4 cube and give estimates up to the 7 7 7 cube through monte carlo methods we find that the number of configurations grows faster with chain length than previously anticipated finally we discuss uniqueness of ground states in the hp model for protein folding

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

the l p regularity problem for the stokes system on lipschitz domains we study the lp regularity problem for the stationary stokes system on lipschitz domains for any p 2 we show that a weak reverse holder inequality with exponent p is both necessary and sufficient for the solvability of the regularity problem with data in w1 p lnp we also obtain the w1 p estimate u lp q lp f lp in a bounded lipschitz domain rd for solutions to the dirichlet problem u q div f div u 0 in and u 0 on where 1p 12 12d e