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

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

the inhomogeneous dirichlet problem forδ2in lipschitz domains abstract we study the inhomogeneous dirichlet problem for the bi laplacian with data given in sobolev and besov spaces on non smooth domains

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

boundary value problems in morrey spaces for elliptic systems on lipschitz domains let q be a bounded lipschitz domain in rn n 3 let be a second order elliptic system with constant coefficients satisfying the legendre hadamard condition we consider the dirichlet problem u 0 in q u f on 4 where e 0 depends on q and 0 a 2 for n 3 we obtain existence and uniqueness results with nontangential maximal function estimate u 2 a q c f 2 a 0q if c satisfies the strong elliptic condition and 0 a min n 1 2 e we show that the neumann type problem u 0 in q g g 2 a dq on 9q vu wfp idci has a unique solution here h2 x dq is an atomic space with the property h2 x dq l2 a 9q the invertibility of layer potentials on l2 x dq and h2 x dq is also obtained finally we study the dirichlet problem for the biharmonic equation we establish a similar estimate in l2 x for the biharmonic equation in which case the range 0 a 2 is sharp for n 4 or 5

boundary value problems for the laplacian in convex and semiconvex domains abstract we study the fully inhomogeneous dirichlet problem for the laplacian in bounded convex domains in r n when the size smoothness of both the data and the solution are measured on scales of besov and triebel lizorkin spaces as a preamble we deal with the dirichlet and regularity problems for harmonic functions in convex domains with optimal nontangential maximal function estimates as a corollary sharp estimates for the green potential are obtained in a variety of contexts including local hardy spaces a substantial part of this analysis applies to bounded semiconvex domains i e lipschitz domains satisfying a uniform exterior ball condition

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

lp polyharmonic dirichlet problems in regular domains iv the upper half space in this article we consider a class of dirichlet problems with lp boundary data for polyharmonic functions in the upper half space by introducing a sequence of new kernel functions for the upper half space called higher order poisson kernels integral representation solutions of the problems are provided

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

closed form solutions to the nonhomogeneous yakubovich transpose matrix equation abstract the nonhomogeneous yakubovich transpose matrix equation x a x t b c y r which contains the well known kalman yakubovich transpose matrix equation and general discrete lyapunov transpose matrix equation as special cases has many important applications in control system theory this study presents two methods to obtain the closed form solutions of the nonhomogeneous yakubovich transpose matrix equation moreover the equivalent forms of the solutions are provided and one of the solutions is established with the controllability matrix the observability matrix and symmetric operator matrix

mean square stability conditions for discrete stochastic bilinear systems necessary and sufficient conditions for mean square stability are proved for the following class of nonlinear dynamical systems finite dimensional bilinear models evolving in discrete time and driven by random sequences the stochastic environment under consideration is characterized only by independence wide sense stationarity and second order properties thus we do not assume random sequences to be gaussian zero mean or ergodic the probability distributions involved are allowed to be arbitrary and unknown limiting state moments are given in terms of the model parameters and disturbances moments

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

on the besov regularity of conformal maps and layer potentials on nonsmooth domains abstract we study the besov regularity of conformal mappings for domains with rough boundary based on the well posedness for the dirichlet problem with besov data also sharp invertibility results for the classical layer potential operators on sobolev besov spaces on the boundary of curvilinear polygons are obtained

a bilinear estimate for biharmonic functions in lipschitz domains we show that a bilinear estimate for biharmonic functions in a lipschitz domain is equivalent to the solvability of the dirichlet problem for the biharmonic equation in as a result we prove that for any given bounded lipschitz domain in mathbb r d and 1 q the solvability of the l q dirichlet problem for 2 u 0 in with boundary data in wa 1 q is equivalent to that of the l p regularity problem for 2 u 0 in with boundary data in wa 2 p where frac 1 p frac 1 q 1 this duality relation together with known results on the dirichlet problem allows us to solve the l p regularity problem for d 4 and p in certain ranges

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

a finite element method for 3d exterior oseen flows error estimates stationary oseen flows in a 3d exterior domain are discretized by applying a p1 p1 stabilized finite element method to the oseen system in a truncated exterior domain with an implicit pointwise artificial boundary condition on the truncating boundary error estimates are proved for this discretization the paper extends an approach which was introduced by guirguis and gunzburger rairo model math anal numer 21 1987 pp 445 464 for exterior stokes flows the stabilized p1 p1 method in question was introduced by rebollo numer math 79 1998 pp 283 319

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

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

integral equation methods for unsteady stokes flow in two dimensions we present an integral equation formulation for the unsteady stokes equations in two dimensions this problem is of interest in its own right as a model for slow viscous flow but perhaps more importantly as an ingredient in the solution of the full incompressible navier stokes equations using the unsteady green s function the velocity evolves analytically as a divergence free vector field this avoids the need for either the solution of coupled field equations as in fully implicit pde based marching schemes or the projection of the velocity field onto a divergence free field at each time step as in operator splitting methods in addition to discussing the analytic properties of the operators that arise in the integral formulation we describe a family of high order accurate numerical schemes and illustrate their performance with several examples

one problem of the navier type for the stokes system in planar domains we study the problem u f u f u g u g in u gu g h h on for a bounded simply connected lipschitz domain in the plane for f 0f 0 g 0g 0 g lp g lp h lq h lq we study a solution in the sense of a nontangential limit for f ws 1 q r2 f ws 1 q r2 g ws q g ws q g wt 1 p p g wt 1 p p h ws 1 q q h ws 1 q q with t s 1t s 1 we prove the existence of a unique solution u wt p r2 ws q u wt p r2 ws q for f bs 1q r r2 g bsq r g bt 1 pp h bs 1 qq r with t s 1t s 1 we prove the existence of a unique solution u btp r2 bsq r for f ck 1 r2 g ck h ck h ck g ck 1 g ck 1 we prove the existence of a unique solution u ck 1 r2 ck

toward solution of matrix equation x af x b c abstract this paper studies the solvability existence of unique solution closed form solution and numerical solution of matrix equation x af x b c with f x x t f x x and f x x h where x is the unknown it is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard stein equations in the form of w a w b c where the dimensions of the coefficient matrices a b and c are the same as those of the original equation closed form solutions of equation x af x b c can then be obtained by utilizing standard results on the standard stein equation on the other hand some generalized stein iterations and accelerated stein iterations are proposed to obtain numerical solutions of equation x af x b c necessary and sufficient conditions are established to guarantee the convergence of the iterations

rank equalities and inequalities for kronecker products of matrices with applications some rank equalities and inequalities are established for kronecker products of matrices as applications upper and lower bounds are presented for the dimensions of the ranges of the two well known linear transformations t 1 x x axb and t 2 x ax xb

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

stability performance and sensitivity analysis of i i d jump linear systems abstractthis paper presents a symmetric kronecker product analysis of independent and identically distributed jump linear systems to develop new lower dimensional equations for the stability and performance analysis of this type of systems than what is currently available in addition new closed form expressions characterising multi parameter relative sensitivity functions for performance metrics are introduced the analysis technique is illustrated with a distributed fault tolerant flight control example where the communication links are allowed to fail randomly

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

consensus over numerosity constrained random networks we analyze the discrete time consensus problem for a group of agents that communicate through a stochastic directed network with fixed out degree this network construction models the perceptual phenomenon of numerosity observed in animal groups exhibiting collective behavior we find necessary and sufficient conditions for mean square consentability of the averaging protocol and we derive a closed form expression for the asymptotic convergence factor analytical results are illustrated through simulations

the structure of shift invariant spaces on a locally compact abelian group abstract we investigate shift invariant subspaces of l 2 g where g is a locally compact abelian group we show among other things that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a parseval frame

on the lid driven problem in a porous cavity a theoretical and numerical approach the purpose of this paper is to analyze the motion of an incompressible viscous fluid through a porous medium located in a two dimensional square cavity i e the lid driven flow problem described by a generalized darcy brinkman model first we study the dirichlet boundary value problem for a generalized darcy brinkman system in a n dimensional bounded lipschitz domain n 2 3 when the given data belongs to some l 2 based sobolev spaces we obtain an existence and uniqueness result for this problem under the assumption of sufficiently small given data an existence and uniqueness result for the robin boundary value problem associated to the same nonlinear system is also formulated further we consider a special dirichlet boundary value problem for the generalized darcy brinkman system i e the lid driven problem associated with such a nonlinear system and we obtain the streamlines of the fluid flow for different reynolds and darcy numbers moreover we consider an additional sliding parameter that corresponds to a mixed dirichlet robin boundary value condition imposed on the upper moving wall and analyze the behavior of the flow with respect to such a parameter

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

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

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

wave diffraction by wedges having arbitrary aperture angle abstract the problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background in fact we may recognize significant research on this topic for more than one century despite this fact up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate sobolev space setting in the present paper we are considering the corresponding boundary value problems for the helmholtz equation with complex wave number admitting combinations of dirichlet and neumann boundary conditions the main ideas are based on a convenient combination of potential representation formulas associated with weighted mellin pseudo differential operators in appropriate sobolev spaces and a detailed fredholm analysis thus we prove that the problems have unique solutions with continuous dependence on the data which are represented by the single and double layer potentials where the densities are solutions of derived pseudo differential equations on the half line

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

curvature flows on surfaces with boundary a theorem of r hamilton 6 and b chow states that for every initial metric the initial value problem 1 possesses a unique solution which is defined for all t 0 moreover for t the solution converges exponentially to a metric of constant gauss curvature different proofs for this result were given by j bartz m struwe and r ye 2 and by m struwe 13 on a surface with boundary one can demand that the gauss curvature in the interior is constant and the geodesic curvature at the boundary vanishes alternatively one can require that the geodesic curvature at the boundary is constant and the gauss curvature in the interior vanishes the evolution equation corresponding to the first case is

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

lipschitz estimates in almost periodic homogenization we establish uniform lipschitz estimates for second order elliptic systems in divergence form with rapidly oscillating almost periodic coefficients we give interior estimates as well as estimates up to the boundary in bounded c1 domains with either dirichlet or neumann data the main results extend those in the periodic setting due to avellaneda and lin for interior and dirichlet boundary estimates and later kenig lin and shen for the neumann boundary conditions in contrast to these papers our arguments are constructive and thus the constants are in principle computable and the results for the neumann conditions are new even in the periodic setting since we can treat nonsymmetric coefficients we also obtain uniform w1 p estimates 2015 wiley periodicals inc

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

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

rank deficient submatrices of kronecker products of fourier matrices abstract we provide a set of maximal rank deficient submatrices of a kronecker product of two matrices a b and in particular the kronecker product of fourier matrices f f n 1 f n k we show how in the latter case maximal rank deficient submatrices can be constructed as tilings of rank one blocks several such tilings may be associated to any subgroup of the abelian group z n 1 z n k that corresponds to the matrix f the maximal rank deficient submatrices of f are also related to an uncertainty principle for fourier transforms over finite abelian groups for which we can then obtain stronger versions

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

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

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

on a conjecture about the norm of lyapunov mappings abstract the norm properties of lyapunov mappings and their restrictions on symmetric and skew symmetric subspaces are investigated for non negative non positive and tridiagonal matrices this paper gives an affirmative answer to an open conjecture which says that the norm of a lyapunov mapping over the space of real square matrices equals that of its restricted map over the subspace of real symmetric matrices in addition matrix transformation and norm relationships on certain matrices arising from the lyapunov mapping and its restrictions are provided

on eigenvalue sets and convergence rate of ito stochastic systems with markovian switching this technical note is concerned with stability analysis and stabilization of ito stochastic systems with markovian switching a couple of eigenvalue sets for some positive operator associated with the stochastic system under study are defined to characterize its stability in the mean square sense properties for these eigenvalue sets are established based on which we show that the spectral abscissa of these eigenvalues sets are the same and thus these eigenvalue sets are equivalent in the sense of characterizing the stability of the system also it is shown that the guaranteed convergence rate of the markovian jump ito stochastic systems can be determined by some eigenvalue set finally a linear matrix inequality based approach is proposed to design controllers such that the closed loop system has guaranteed convergence rate some numerical examples are carried out to illustrate the effectiveness of the proposed approach the research in this technical note opens several perspectives for future work stated as some open problems

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

large solutions for yamabe and similar problems on domains in riemannian manifolds we present a unifled approach to study large positive solutions i e u x 1 as x of the equation u hu i k u if in an arbitrary domain we assume u is convex and grows su ciently fast as u 1 equations of this type arise in geometry yamabe problem two dimensional curvature equation probability superdifiusion we prove that both existence and uniqueness are local properties of points of the boundary i e they depend only on properties of in arbitrary small neighborhood of each boundary point we also flnd several new necessary and su cient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries

spectral properties of sums of certain kronecker products we introduce two kinds of sums of kronecker products and their induced operators we study the algebraic properties of these two kinds of matrices and their associated operators the properties include their eigenvalues their eigenvectors and the relationships between their spectral radii or spectral abscissae furthermore two projected matrices of these kronecker products and their induced operators are also studied

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

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

kronecker maps and sylvester polynomial matrix equations the concepts of kronecker maps and sylvester polynomial matrix equations are proposed some properties of the kronecker maps are deduced it is shown that many kinds of matrix equations can be formulated into the so called sylvester polynomial matrix equations with the help of the kronecker maps general explicit solutions of the sylvester polynomial matrix equations are established when the polynomial matrices possess f primeness for the nonsquare case the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix

a posteriori error estimates of stabilized low order mixed finite elements for the stokes eigenvalue problem in this paper we obtain a priori and a posteriori error estimates for stabilized low order mixed finite element methods for the stokes eigenvalue problem we prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove that up to higher order terms the estimator is equivalent to the energy norm of the error we also present some numerical tests which show the performance of the adaptive scheme

analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators we deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle we assume that the boundary of the obstacle is a non analytic hypersurface we show that when we put dirichlet boundary conditions one measurement is enough to reconstruct the obstacle in the neumann case we have results only for n 2 3 in general more precisely we show that one measurement is enough for n 2 and we need 3 linearly independent inputs for n 3 however in the case for the helmholtz equation we only need n 1 linearly independent inputs for any n 2 here n is the dimension of the space containing the obstacle these are justified by investigating the analyticity properties of the zero set of a real analytic function in addition we give a reconstruction procedure for each case to recover the shape of obstacle although we state the results for the scattering problems similar results are true for the associated boundary value problems

a free boundary problem for a hamilton jacobi equation arising in ion etching in this paper we study a model arising in the fabrication of a semiconductor device by ion etching in this model we have two different materials with their respective sputtering functions fi i 1 2 one material is on the top of the other as shown in fig 1 and their common material boundary is given by y g x let us denote by i the region in the x t plane where the upper material is located and by ii the region in the x t plane where the

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

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

estimates for the deviation of solutions and eigenfunctions of second order elliptic dirichlet boundary value problems under domain perturbation estimates in suitable lebesgue or sobolev norms for the deviation of solutions and eigenfunctions of second order uniformly elliptic dirichlet boundary value problems subject to domain perturbation in terms of natural distances between the domains are given the main estimates are formulated via certain natural and easily computable atlas distances for domains with lipschitz continuous boundaries as a corollary similar estimates in terms of more classical distances such as the hausdorff distance or the lebesgue measure of the symmetric difference of domains are derived sharper estimates are also proved to hold in smoother classes of domains

on h2 model reduction of bilinear systems the h 2 model reduction problem for continuous time bilinear systems is studied in this paper by defining the h 2 norm of bilinear systems in terms of the state space matrices the h 2 model reduction error is computed via the reachability or observability gramian necessary conditions for the reduced order bilinear models to be h 2 optimal are given the gradient flow approach is used to obtain the solution of the h 2 model reduction problem the formulation allows certain properties of the original models to be preserved in the reduced order models the model reduction procedure developed can also be applied to finite dimensional linear time invariant systems a numerical example is employed to illustrate the effectiveness of the proposed method

the stationary navier stokes system in nonsmooth manifolds the poisson problem in lipschitz and c 1 domains we consider the linearized version of the stationary navier stokes equations on a subdomain of a smooth compact riemannian manifold m the emphasis is on regularity the boundary of is assumed to be only c1 and even lipschitz and the data are selected from appropriate sobolev besov scales our approach relies on the method of boundary integral equations suitably adapted to the variable coefficient setting we are considering here applications to the stationary nonlinear navier stokes equations in this context are also discussed

boundary korn inequality and neumann problems in homogenization of systems of elasticity this paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization we establish uniform optimal regularity estimates for solutions of neumann problems in a bounded lipschitz domain with l2 boundary data the proof relies on a boundary korn inequality for solutions of systems of linear elasticity and uses a large scale rellich estimate obtained in shen anal pde arxiv 1505 00694v2

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

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

stationary stokes oseen and navier stokes equations with singular data the concept of very weak solution introduced by giga math z 178 287 329 1981 for the stokes equations has hardly been studied in recent years for either the navier stokes equations or the navier stokes type equations we treat the stationary stokes oseen and navier stokes systems in the case of a bounded open set connected of class mathcal c 1 1 of mathbb r 3 taking up once again the duality method introduced by lions and magenes problemes aus limites non homogenes et applications vols 1 2 dunod paris 1968 and giga math z 178 287 329 1981 for open sets of class mathcal c infty see also chapter 4 of necas les methodes directes en theorie des equations elliptiques french masson et cie ed paris academia editeurs prague 1967 which considers the hilbertian case p 2 for general elliptic operators we give a simpler proof of the existence of a very weak solution for stationary oseen and navier stokes equations when data are not regular enough based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields in the stationary navier stokes case the results will be valid for external forces not necessarily small which lets us extend the uniqueness class of solutions for these equations considering more regular data regularity results in fractional sobolev spaces will also be discussed for the three systems all these results can be extended to other dimensions

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

on discrete stochastic bilinear systems stability abstract mean square stability conditions for discrete time bilinear systems operating in a stochastic environment are given in this paper only independence and wide sense stationarity are required for the second order disturbance sequences involved thus dismissing ergodicity and zero mean assumptions stochastic stability conditions are derived by using a deterministic stability result for a class of separable nonlinear dynamical systems evolving in a banach space

mixed boundary value problems for the stokes system we prove the well posedness of the mixed problem for the stokes system in a class of lipschitz domains in n n 3 the strategy is to reduce the original problem to a boundary integral equation and we establish certain new rellich type estimates which imply that the intervening boundary integral operator is semi fredholm we then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the lame system which has recently been shown to be invertible

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

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

on gramians and balanced truncation of discrete time bilinear systems this paper obtains a balanced truncation model reduction method for discrete time bilinear systems after deriving the sufficient conditions for the discrete time bilinear systems to lead to bounded outputs for some bounded inputs we study the properties of the reachability and observability gramians the gramians can be computed by solving two generalized lyapunov equations by balancing the reachability and observability gramians a model reduction method is obtained some propositions of the reduced order models are given a numerical example is given to illustrate the effectiveness of the proposed method

the lp dirichlet problem for second order elliptic operators and a p adapted square function abstract we establish l p solvability for 1 p of the dirichlet problem on lipschitz domains with small lipschitz constants for elliptic divergence and non divergence type operators with rough coefficients obeying a certain carleson condition with small norm