intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators abstract we consider a one dimensional generalized diffusion operator g on an open interval which is not necessarily bounded we characterize sp i e 1 is a small perturbation of the operator g in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for iu i e the associated heat kernel is intrinsically ultracontractive

on the modelling of incompressibility in linear and non linear elasticity with the master slave approach this is the pre peer reviewed version of the following article munoz jose j on the modelling of incompressibility in linear and non linear elasticity with the master slave approach international journal for numerical methods in engineering abril 2008 vol 74 num 2 p 269 293 which has been published in final form at http www3 interscience wiley com journal 116323442 abstract

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

general gauge and conditional gauge theorems general gauge and conditional gauge theorems are established for a large class of not necessarily symmetric strong markov processes including brownian motions with singular drifts and symmetric stable processes furthermore new classes of functions are introduced under which the general gauge and conditional gauge theorems hold these classes are larger than the classical kato class when the process is brownian motion in a bounded c 1 1 domain

analytic perturbation of generalized inverses abstract we investigate the analytic perturbation of generalized inverses firstly we analyze the analytic perturbation of the drazin generalized inverse also known as reduced resolvent in operator theory our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent it allows us to treat regular and singular perturbations in a unified framework we provide an algorithm for computing the coefficients of the laurent series of the perturbed drazin generalized inverse in particular the regular part coefficients can be efficiently calculated by recursive formulae finally we apply the obtained results to the perturbation analysis of the moore penrose generalized inverse in the real domain

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

an estimate of the babuska brezzi inf sup discrete stability constant for general linear petrov galerkin finite element formulations an estimate of the babuska brezzi stability constant an a priori lower bound estimate of the babuska brezzi bb inf sup discrete stability constant is given for general linear petrov galerkin formulations the estimate is based on reasonable assumptions regarding a specific auxiliary adjoint problem the similarities and discrepancies of the presented method compared to the well known fortin s criterion are also noted the estimate may be used to obtain numerical verification of the bb condition under the assumption that the finite element basis functions are almost orthogonal

perturbation of spectral inverses applied to a boundary layer phenomenon arising in chemical networks abstract a linear model showing behavior of the boundary layer type is developed the governing equations have the form t b t x t 0 t 0 where the coefficient matrix b depends continuously on two nonnegative parameters and has a discontinuity in rank at the origin of the parameter space the analysis depends on a regularization technique for linear systems and on a perturbation result for spectral group generalized inverses

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

intrinsic ultracontractivity of nonsymmetric diffusions with measure valued drifts and potentials recently in preprint 2006 we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups in this paper we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure valued drifts and measure valued potentials in bounded domains our process y is a diffusion process whose generator can be formally written as l with dirichlet boundary conditions where l is a uniformly elliptic second order differential operator and 1 d is such that each component i i 1 d is a signed measure belonging to the kato class k d and v is a nonnegative measure belonging to the kato class k d 2 we show that scale invariant parabolic and elliptic harnack inequalities are valid for y in this paper we prove the parabolic boundary harnack principle and the intrinsic ultracontractivity for the killed diffusion y d with measure valued drift and potential when d is one of the following types of bounded domains twisted holder domains of order a e 1 3 1 uniformly holder domains of order a 0 2 and domains which can be locally represented as the region above the graph of a function this extends the results in j funct anal 100 1991 181 206 and probab theory related fields 91 1992 405 443 as a consequence of the intrinsic ultracontractivity we get that the supremum of the expected conditional lifetimes of y d is finite

brownian motion in cones we study the asymptotic behavior of brownian motion and its conditioned process in cones using an infinite series representation of its transition density a concise probabilistic interpretation of this series in terms of the skew product decomposition of brownian motion is derived and used to show properties of the transition density

the patch test for mixed formulations this paper presents a simple extension of the patch test to mixed formulations to provide the necessary and sufficient conditions for convergence the general algebraic conditions of babuska and brezzi are given a simple form and a conceptual application of the patch tests serves to point out the instability of several well known formulations for incompressible problems the test is also applied to mixed displacement strain formulations of elasticity and limiting formulations are noted

the lifetime of conditioned brownian motion in certain lipschitz domains for certain lipschitz domains d we obtain a series expansion for the distribution of the lifetime d of conditioned brownian motion on d from this we determine mathop lim limits t to infty text t 1 text log text p x h tau d text t mathop lim limits t to infty text t 1 text log text p x tau d text t text lambda d text where d is the first eigenvalue of 1 2 on d

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

finite element methods principles for their selection abstract principles for the selection of a finite element method for a particular problem are discussed these principles are stated in terms of the notion of approximability optimality and stability several examples are discussed in detail as illustrations conclusions regarding the selection of finite element methods are summarized in the final section of the paper

progressive intrinsic ultracontractivity and heat kernel estimates for non local schrodinger operators abstract we study the long time asymptotic behaviour of semigroups generated by non local schrodinger operators of the form h l v the free operator l is the generator of a symmetric levy process in r d d 1 with non degenerate jump measure and v is a sufficiently regular confining potential we establish sharp two sided estimates of the corresponding heat kernels for large times and identify a new general regularity property which we call progressive intrinsic ultracontractivity to describe the large time evolution of the corresponding schrodinger semigroup we discuss various examples and applications of these estimates for instance we characterize the heat trace and heat content our examples cover a wide range of processes and we have to assume only mild restrictions on the growth resp decay of the potential and the jump intensity of the free process our approach is based on a combination of probabilistic and analytic methods our examples include fractional and quasi relativistic schrodinger operators

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

a novel unsymmetric 8 node plane element immune to mesh distortion under a quadratic displacement field an 8 node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions viz the compatibility enforcing shape functions and completeness enforcing shape functions the former are chosen to satisfy exactly the minimum inter as well as intra element displacement continuity requirements while the latter are chosen to satisfy all the linear and higher order completeness requirements so as to reproduce exactly a quadratic displacement field numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions in this respect the proposed 8 node unsymmetric element emerges to be better than the existing symmetric quad8 quad8 9 quad9 quad12 and quad16 elements and matches the performance of the quartic element quad25 for test problems involving a cubic or higher order displacement field the proposed element yields a solution accuracy that is comparable to or better than that of quad8 quad8 9 and quad9 elements furthermore the element maintains a good accuracy even with the reduced 2 2 numerical integration copyright 2003 john wiley sons ltd

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

uniformly fat sets in this paper we study closed sets e which are locally uniformly fat with respect to a certain nonlinear riesz capacity we show that e is actually locally uniformly fat with respect to a weaker riesz capacity two applications of this result are given the first application is concerned with proving sobolev type inequalities in domains whose complements are uniformly fat the second application is concerned with the fekete points of e introduction let x x x2 x be a point in euclidean n space rn with jxl the norm of x for a 0 let i x ixi a n denote the riesz kernel of order a and put i f x f ix y f n f y dy x e r rn ic u x fr yj c n d u y x e r rn when f is a lebesgue integrable function and u a borel measure on rn here dy denotes lebesgue measure on rn if e c rn 0 1 define the a p outer riesz capacity of e by ra p e inf jfllfp ia f 1 on el where if lip is the lebesgue p norm of f we note that a 1 p 2 is the classical newtonian capacity if ap n we let rc p e inf jfjjf j f 1 on el where jc is the truncated riesz kernel defined by je x ixi a n 100 a n x 100 0 lxl 100 we shall say that a property holds a p quasi everywhere abbreviated a p q e on a set e if it holds on e except perhaps for a set of rop capacity zero next we list some properties of riesz capacities to simplify the writing we state these properties only for 0 u l p l 1 a p q e on e and p 1 a p q e on supp u c e moreover e jjli pi ujj rop e received by the editors march 11 1987 1980 mathematic3 subject cla33ification 1985 revi3ion primary 31c15 31b15 1988 american mathematical society 0002 9947 88 1 00 25 per page

distoriton of area and conditioned brownian motion in a simply connected planar domaind the expected lifetime of conditioned brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain we show that each hyperbolic geodesic induces a decomposition ofd into disjoint subregions omega j mathop cup limits j omega j d and that the subregions are obtained in a natural way using euclidean geometric quantities relating tod the lifetime associated with on each j is then shown to be bounded by the product of the diameter of the smallest ball containing j and the diameter of the largest ball in j because this quantity is never larger than and in general is much smaller than the area of the largest ball in j it leads to finite lifetime estimates in a variety of domains of infinite area

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

norm estimate of green operator perturbation of green function and integrability of superharmonic functions an operator norm estimate of the green operator for the laplace equation on an open set in the euclidean space is given in terms of capacity this applies to the perturbation of the green function for the schrodinger equation a sufficient condition for the green function for the schrodinger equation to have decay near the boundary comparable to that for the laplace equation is given in terms of an integrable condition for the potential of the schrodinger equation this has an application to the martin boundary of a smooth domain with respect to the schrodinger equation also it is related to the cranston mcconnell inequality about the life time estimate and the integrability of nonnegative superharmonic functions 1991 mathematics subject classification 31a35 31b35 31c35

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

iterative aggregation disaggregation method for computing stationary probability vectors of markov type operators a global convergence of a class of iterative aggregation disaggregation methods is presented and some applications as well as some conclusions of numerical experiments performed are shown

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

the babuska brezzi condition and the patch test an example abstract this paper discusses an example of a mixed finite element method in one dimension which satisfies the engineering patch test but does not satisfy a uniform inf sup condition the inf sup constant for this method is shown to be of order h2 the reasons for good results from benchmark problems for this method are addressed by characterizing the nature of input data for which the method performs well numerical results illustrating the effect of the non uniform inf sup condition are presented

holder domains and the boundary harnack principle the constant c depends on l only through the constant cl defined in 1 below of course c also depends on d v and k the above result shows that the boundary harnack principle holds in all holder domains it was first proven for the range 1 2 1 by bass and burdzy 1990a theorems 3 5 and 4 5 it was subsequently extended by banuelos unpublished to the full range of provided the domain also satisfies a uniform capacity condition on the boundary in banuelos 1990 a class of domains called uniformly holder domains of order was introduced a boundary harnack principle may be obtained for such domains for all 0 1 by a variation of the proof of theorem 1 in order to make this note compact we refer

on an estimate of cranston and mcconnell for elliptic diffusions in uniform domains we show that ifd n n 3 n 3 is a bounded uniform domain then the lifetime of the doobh paths ind for elliptic diffusions in divergence form is finite this result holds for any bounded domaind in the plane

a nonlinear model involving m operators an amplification effect measured in the cascade of vision an amplification effect measured in the cascade of vision forms a basis to formulate an abstract nonlinear model offering a generalization of the cascade problem this model is analyzed and some conditions concerning the existence and or possible uniqueness and multiplicity of solutions are presented an essential feature of the model is that the appropriate operators governing the problem are not surjective for the case of unique solvability a numerical procedure is proposed and its convergence is proved

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

a model of amplification abstract a linear model showing behavior of the boundary layer type is developed as a generalization of such a matrix model of bohl and lancaster 1993 the governing equations have the form x t bx t x t 0 t 0 where the operator b being the infinitesimal generator of a semigroup of operators of class c 0 depends continuously on a parameter and has discontinuity in the dimension of the null space of b it is shown that a singular behavior can be detected by a single parameter of imperfection the analysis depends on a regularization technique for linear operator systems and on a perturbation result of spectral generalized inverse drazin a partial order of the appropriate banach spaces is another important tool in the investigation

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

the finite element patch test revisited a computer test for convergence validation and error estimates abstract the patch test was introduced in the mid 1960s first as a check of satisfactory performance for elements violating continuity requirements and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors in its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct later it was established that the test could also be used to find the convergence order of any finite element type while consistency is one of the necessary requirements for convergence the second is of course that of stability in the original early work only displacement type structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms in a 1986 reformulation of the patch test this stability or solvability requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms 1 2 although some misinterpretation of the patch test has occurred in literature 3 4 the authors believe that with certain restrictions on regularity of the problem dealt with the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical babuska brezzi criteria 5 6 in recent years another important use of the patch test has been found by babuska and his coworkers 7 9 this work tests the asymptotic effectivity of various error estimating procedures used with various element patterns