on the l r hodge theory in complete non compact riemannian manifolds we study solutions for the hodge laplace equation delta u omega on p forms with displaystyle l r estimates for displaystyle r 1 our main hypothesis is that delta has a spectral gap in displaystyle l 2 we use this to get sl non classical displaystyle l r hodge decomposition theorems an interesting feature is that to prove these decompositions we never use the boundedness of the riesz transforms in displaystyle l s par these results are based on a generalisation of the raising steps method to complete non compact riemannian manifolds

use of higher order shape functions in the scaled boundary finite element method the scaled boundary finite element method is a novel semi analytical technique whose versatility accuracy and efficiency are not only equal to but potentially better than the finite element method and the boundary element method for certain problems this paper investigates the possibility of using higher order polynomial functions for the shape functions two techniques for generating the higher order shape functions are investigated in the first the spectral element approach is used with lagrange interpolation functions in the second hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones as in the p version of the finite element method to check the accuracy of the proposed procedures a plane strain problem for which an exact solution is available is employed a more complex example involving three scaled boundary subdomains is also addressed the rates of convergence of these examples under p refinement are compared with the corresponding rates of convergence achieved when uniform h refinement is used allowing direct comparison of the computational cost of the two approaches the results show that it is advantageous to use higher order elements and that higher rates of convergence can be obtained using p refinement instead of h refinement copyright 2005 john wiley sons ltd

inequalities for two type potential operators on differential forms abstract the purpose of this paper is to derive some strong type inequalities for convolution type potential operator applied on differential forms caccioppoli type inequalities for integral type potential operator acting on a harmonic tensor are also obtained

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

hybrid uncertainty analysis of functionally graded plates via multiple imprecise random field modelling of uncertain material properties abstract this paper presents the mixed probabilistic non probabilistic uncertainty problem regarding the hybrid uncertainty propagation on physical model of functionally graded plates through dissimilarly uncertain characterisation of material properties in the presence of the multiple imprecise random field modelling the spectral stochastic finite element framework is extended to perform mixed uncertainty analysis of functionally graded material whereby the spatial variation of elastic modulus of the plate is uniquely characterised as the multiple imprecise random field modelling more than a purely homogeneous random field this characterisation has originated in heeding the construction of physical models and the significant influences of separately uncertain characteristics propagated on the physical model through different material ingredients are examined to make physical models more reliable and realistic mechanics and mathematics intrusive formulations are then established to explicitly produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation bounded surfaces of statistical moments and bounding distribution functions are vividly depicted for demonstration the feasibility and effectiveness of the presented work are illustrated via test cases of numerical examples

higher integrability of iterated operators on differential forms abstract in this paper we first prove the local higher integrability and higher order imbedding theorems for the iterated operators defined on differential forms then we prove the global higher integrability and higher order imbedding inequalities for these operators finally we demonstrate applications of the main results by examples

norms of the composition of the maximal and projection operators abstract in this paper we develop different norm estimates for the compositions of the maximal and projection operators applied to the solutions of a nonlinear elliptic partial differential equation as applications we also obtain some estimates related to the jacobian and exponents of mappings

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

a nurbs based hybrid collocation galerkin method for the analysis of boundary represented solids abstract the paper is concerned with a new numerical method nurbs based hybrid collocation galerkin method nurbs hcgm to solve the in plane motion problem of elastic solids it combines the merits of the so called scaled boundary finite element method sb fem and the isogeometric collocation method for the analysis the boundary scaling technique of sb fem is adopted it leads to a formulation where only the boundary of a structure is discretized here the nurbs basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary this is in accordance with the boundary representation modeling technique which is commonly employed in computer aided design software the inner domain is described by a radial scaling parameter applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation ode of euler type where the unknown displacements are a function of the radial scaling parameter in the present work a nurbs based collocation scheme is introduced to solve this equation nurbs basis functions are suggested for the approximation of the displacements in scaling direction the higher continuity provided by nurbs allows to use collocation to solve the ode directly instead of using the weak form in scaling direction the proposed approach is validated by comparison with the eigenvalue solution of the ode it is remarked that the eigenvalue solution is restricted to linear problems whereas the proposed method could be extended to nonlinear problems in general the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries the accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions the computational cost is investigated with the help of numerical examples and is compared to isogeometric galerkin approach

biharmonic maps from r 4 into a riemannian manifold for a domain r 4 and a compact riemannian manifold n r k without boundary if u w 2 2 n is an extrinsic or intrinsic respectively biharmonic map then u c n

moving boundary vortices for a thin film limit in micromagnetics we study the limiting behavior of solutions of the landau lifshitz gilbert equation belonging to thin films of ferromagnetic materials in the appropriate time scale and under reasonable conditions there is a subsequence converging to a map that has vortices at two boundary points the vortices move holder continuously in time and the map satisfies a formal euler lagrange equation away from the vortices 2004 wiley periodicals inc

stochastic finite element analysis of structures in the presence of multiple imprecise random field parameters the research work presents the study on non deterministic problems in the presence of the multiple imprecise random field uncertainties by extending the spectral stochastic finite element framework the standard random field is expanded to characterize the behaviour of physical model with imprecise randomness more appropriately for real engineering problems based on available sources of uncertainty the young s modulus and body force of structures are considered as imprecise random fields with bounded statistical moments mathematical expressions and solution procedure are developed to produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation and bounding distribution functions the bounding probability density and cumulative distribution of the imprecise random response are then effectively visualized by means of polynomial chaos expansion the feasibility and effectiveness of the presented method are illustrated by three numerical examples

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

global weighted inequalities for operators and harmonic forms on manifolds abstract in this paper both the local and global weighted sobolev poincare imbedding inequalities and poincare inequalities for the composition t g are established where t is the homotopy operator and g is green s operator applied to a harmonic forms on manifolds

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

erratum to on the weak l p hodge decomposition and beurling ahlfors transforms on complete riemannian manifolds based on a new martingale representation formula we prove some quantitative upper bound estimates of the l p norm of some singular integral operators on complete riemannian manifolds this leads us to establish the weak l p hodge decomposition theorem and to prove the l p boundedness of the beurling ahlfors transforms on complete non compact riemannian manifolds with non negative weitzenbock curvature operator

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

qualitative properties of multi bubble solutions for nonlinear elliptic equations involving critical exponents abstract the objective of this paper is to obtain qualitative characteristics of multi bubble solutions to the lane emden fowler equations with slightly subcritical exponents given any dimension n 3 by examining the linearized problem at each m bubble solution we provide a number of estimates on the first n 2 m eigenvalues and their corresponding eigenfunctions specifically we present a new and unified proof of the classical theorems due to bahri li rey 1995 2 and rey 1999 24 which state that if n 4 or n 3 respectively then the morse index of a multi bubble solution is governed by a certain symmetric matrix whose component consists of a combination of green s function the robin function and their first and second derivatives

representation of singular fields without asymptotic enrichment in the extended finite element method summary in this paper we replace the asymptotic enrichments around the crack tip in the extended finite element method xfem with the semi analytical solution obtained by the scaled boundary finite element method sbfem the proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the xfem to model cracks in homogeneous and or heterogeneous materials without a priori knowledge of the asymptotic solutions a heaviside enrichment is used to represent the jump across the discontinuity surface we call the method as the extended sbfem numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number a simple code is annexed to compute the terms in the stiffness matrix which can easily be integrated in any existing fem xfem code copyright 2013 john wiley sons ltd

slow mappings of finite distortion we examine mappings of finite distortion from euclidean spaces into riemannian manifolds we use integral type isoperimetric inequalities to obtain liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds

an adaptive spectral galerkin stochastic finite element method using variability response functions a methodology is proposed in this paper to construct an adaptive sparse polynomial chaos pc expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields the proposed methodology utilizes the concept of variability response function vrf in order to compute an a priori low cost estimate of the spatial distribution of the second order error of the response as a function of the number of terms used in the truncated karhunen lo eve kl expansion this way the influence of the response variance to the spectral content correlation structure of the random input is taken into account through a spatial variation of the truncated kl terms the criterion for selecting the number of kl terms at different parts of the structure is the uniformity of the spatial distribution of the second order error this way a significantly reduced number of pc coefficients with respect to classical pc expansion is required in order to reach a uniformly distributed target second order error this results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method ssfem copyright c 2014 john wiley sons ltd

convex functionals and the stratification of the singular set of their stationary points abstract we prove partial regularity of stationary solutions and minimizers u from a set r n to a riemannian manifold n for the functional f x u u 2 d x the integrand f is convex and satisfies some ellipticity and boundedness assumptions we also develop a new monotonicity formula and an regularity theorem for such stationary solutions with no restriction on their images we then use the idea of quantitative stratification to show that the k th strata of the singular set of such solutions are k rectifiable

consistent infinitesimal finite element cell method three dimensional vector wave equation to calculate the unit impulse response matrix of an unbounded medium for use in a time domain analysis of unbounded medium structure interaction the consistent infinitesimal finite element cell method is developed for the three dimensional vector wave equation this is a boundary finite element procedure the discretization is only performed on the structure medium interface yielding a reduction of the spatial dimension by 1 the procedure is rigorous in the radial direction and exact in the finite element sense in the circumferential directions in contrast to the boundary element procedure the consistent infinitesimal finite element cell method does not require a fundamental solution and incorporates interfaces extending from the structure medium interface to infinity compatible with similarity without any additional computational effort a general anisotropic material can be processed the derivation is based on the finite element formulation and on similarity

three dimensional stochastic finite element method for elasto plastic bodies a new stochastic finite element method sfem is formulated for three dimensional softening elasto plastic bodies with random material properties the method is based on the karhunen loeve and polynomial chaos expansions and able to efficiently estimate complete probabilistic characteristics of the response such as moments or pdfs to reduce the computational complexity in the three dimensional setting two alterations are made with respect to the two dimensional sfem proposed earlier by the authors first a variability preserving modification of the karhunen loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties second an efficient algorithm for parallel processing is developed with time consumption being the same order as for an ordinary fem rendering the proposed sfem an effective alternative to monte carlo simulation the applicability of the proposed method to stochastic analysis of strain localization is examined using monte carlo simulation then it is applied to a fault formation problem which is a recent concern of earthquake engineering ground surface layers are modelled by a softening elasto plastic body and the evolution of probabilistic characteristics of the rupture process is analysed in detail some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint copyright 2001 john wiley sons ltd

ls μ averaging domains abstract in this paper we first introduce l s averaging domains which are generalizations of l s averaging domains introduced by s g staples we characterize l s averaging domains using the quasihyperbolic metric as applications we prove norm inequalities for conjugate a harmonic tensors in l s averaging domains which can be considered as generalizations of the hardy and littlewood theorem for conjugate harmonic functions finally we give applications to quasiconformal and quasiregular mappings

weighted poincare type inequalities for differential forms inls μ averaging domains abstract we first introduce a new class of weighted functions and obtain some basic properties of this class then as applications of this class we prove local weighted integral inequalities for differential forms finally we obtain the global weighted integral inequalities for differential forms inls averaging domains which can be considered as generalizations of the poincare inequality for sobolev functions

an efficient forward propagation of multiple random fields using a stochastic galerkin scaled boundary finite element method abstract this paper serves to extend the existing literature on the stochastic galerkin scaled boundary finite element method sgsbfem in two ways the first part of this work deals with the formulation of multiple non correlated gaussian random fields using the conventional karhunen loeve expansion technique and its forward propagation through the spectral stochastic scaled boundary finite element setting using the polynomial surface fit method in terms of the scaled boundary coordinates the advantages in adopting such a forward propagation technique in capturing the statistical moments of quantities of interest qoi across the domain are highlighted using carefully chosen linear elastic problems having large to least correlated random fields as inputs the second contribution is the extension of the proposed forward uncertainty quantification uq to take into account multiple independent random fields followed by polynomial chaos expansion pce based sensitivity analysis both the computational efficiency and the accuracy of the proposed framework under different input random field correlation settings are elaborated upon by comparing their results against that obtained using the current existing sgsbfem in the literature moreover the stochastic results are validated for all the numerical examples using the monte carlo method

strain and stress computations in stochastic finite element methods this paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element method based on the polynomial chaos expansion it identifies the cases where this objective can be achieved by analytical means using the orthogonality property of the chaos polynomials and those where it requires a numerical integration technique to this effect the applicability and efficiency of several numerical integration schemes are considered these include the gauss hermite quadrature with the direct tensor product also known as the kronecker product smolyak s approximation of such a tensor product monte carlo sampling and the latin hypercube sampling method an algorithm for reducing the dimensionality of integration under a direct tensor product is also explored for optimizing the computational cost and complexity the convergence rate and algorithmic complexity of all of these methods are discussed and illustrated with the non deterministic linear stress analysis of a plate copyright 2007 john wiley sons ltd

boundary vortices for thin ferromagnetic films we consider a simplified version of the micromagnetic energy for ferromagnetic samples in the shape of thin films we study a stationary stable critical points and b solutions of the corresponding landau lifshitz equation under a stability condition we determine the asymptotic behaviour of solutions of these variational problems in the thin film limit a characteristic property of the limit is the development of ginzburg landau type vortices at the boundary

inequalities for green s operator with lipschitz and bmo norms in this paper we obtain lipschitz norm and bmo norm inequalities for green s operator applied to differential forms we also establish norm comparison theorems for the bmo norm and the lipschitz norm of green s operator

inequalities for hilbert operator and its extensions the probabilistic approach we present a probabilistic study of the hilbert operator tf x 1 0f y dyx y x 0 tf x 1 0 f y dyx y x 0 defined on integrable functions ff on the positive halfline using appropriate novel estimates for orthogonal martingales satisfying the differential subordination we establish sharp moment weak type and inequalities for tt we also show similar estimates for higher dimensional analogues of the hilbert operator and by the further careful modification of martingale methods we obtain related sharp localized inequalities for hilbert and riesz transforms

weighted norm inequalities for conjugatea harmonic tensors abstract we prove weighted normal inequalities for conjugate a harmonic tensors in john domains which can be considered as generalizations of the hardy and littlewood theorem for conjugate harmonic functions

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

invariance properties of lϕ μ averaging domains under some mappings abstract in this paper we first show that l averaging domains are invariant under some mappings such as k quasi isometries and quasi isometries then we prove that john domains are l averaging domains for any satisfying the conditions in the definition of l averaging domains

the stochastic galerkin scaled boundary finite element method on random domain summary the research work extends the scaled boundary finite element method sbfem to non deterministic framework defined on random domain wherein random behaviour is exhibited in the presence of the random field uncertainties the aim is to blend the sbfem into the galerkin spectral stochastic methods which leads to a proficient procedure for handling the stress singularity problems and crack analysis the young s modulus of structures is considered to have random field uncertainty resulting in the stochastic behaviour of responses mathematical expressions and the solution procedure are derived to evaluate the statistical characteristics of responses displacement strain and stress and stress intensity factors of cracked structures the feasibility and effectiveness of the presented method are demonstrated by particularly chosen numerical examples this article is protected by copyright all rights reserved

numerical solution of spectral stochastic finite element systems abstract this paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element ssfem formulation two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems a brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure the differences between these matrices from their deterministic finite element counterparts are illustrated an iterative solution scheme is proposed which utilizes their specific properties for efficient memory management and enhanced convergence behavior results from numerical tests are presented comparisons with standard algorithms illustrate the efficiency of the proposed algorithm the second solution procedure presented in this paper is based on hierarchical basis concepts results from numerical tests are again provided and the limitations of this approach are assessed the performance of both proposed algorithms indicates that the linear algebraic systems from the underlying ssfem formulation can be solved with considerably less effort in memory and computation time than their size suggests furthermore the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic fem

the bird s eye view on finite element method for structures with large stochastic variations abstract extensive motivation to do an additional work in the finite element method in stochastic problems femsp is discussed the qualitative comparison of femsp with the state of the art of the deterministic fem is given these observations and thoughts are then realized in several manners we first present the exact inverse femsp which however cannot serve as a general tool for stochastic analysis of complex structures the new variational principles for stochastic beams for the mean response function as well as response s auto correlation function are formulated and the fem based on the variational principles is presented finally a general non perturbative fem for stochastic problems with large variations is developed based on the element level flexibility it is concluded that much work needs to be done in order femsp to be at the level compared to that of the deterministic fem the advantage of the main methods presented here over the conventional ones lies in their non perturbational nature numerical examples are presented

on weak continuity and the hodge decomposition we address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first order differential constraints we prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives this is sufficient for reproducing the div curl lemma of murat and tartar the null lagrangians in the calculus of variations and the weakly continuous polynomials for maxwell s equations this result was derived independently by tartar who stated it in a recent survey article 7 our proof is explicit and uses the hodge decomposition

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

mappings of finite distortion the degree of regularity this paper investigates the self improving integrability properties of the so called mappings of finite distortion let k x 1 be a measurable function defined on a domain rn n 2 and such that exp k x lloc1 0 we show that there exist two universal constants c1 n c2 n with the following property let f be a mapping in wloc1 1 rn with df x n k x j x f for a e x and such that the jacobian determinant j x f is locally in l1log c1 n l then automatically j x f is locally in l1logc2 n l this result constitutes the appropriate analog for the self improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion namely we obtain novel results on the size of removable singularities for bounded mappings of finite distortion and on the area distortion under this class of mappings

partial regularity for stationary harmonic maps at a free boundary let and be smooth riemannian manifolds of the dimension n 2 with nonempty boundary and compact without boundary we consider stationary harmonic maps u h 1 with a free boundary condition of the type u given a submanifold we prove partial boundary regularity namely sing u 0 a result that was until now only known in the interior of the domain see b the key of the proof is a new lemma that allows an extension of u by a reflection construction once the partial regularity theorem is known it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold

on the computation of mixed mode stress intensity factors in functionally graded materials a new interaction energy integral method for the computation of mixed mode stress intensity factors at the tips of arbitrarily oriented cracks in functionally graded materials is described in the method interaction energy contour integrals are defined and expressed in equivalent domain form the interaction energy integrals involve products of the actual fields that arise from solution to the boundary value problem with known auxiliary fields the auxiliary stress and displacement fields are chosen to be the asymptotic near tip fields for a crack in a homogeneous material having the same elastic constants as those found at the crack tip in the functionally graded material the auxiliary strain fields are obtained from the auxiliary stress fields using the constitutive relation for the functionally graded material a consequence of our choice for the auxiliary strain field is lack of compatibility which leads to extra terms in the domain integrals that need to be evaluated for the sake of accuracy the mixed mode stress intensity factors are obtained from the domain integrals as a post processing step in the extended finite element method to assess the accuracy of the method we consider the benchmark problems of an edge cracked plate and an angled center crack for specimens with a functional gradient in material properties excellent agreement is obtained between the numerical results and the analytical solutions for both stress intensity factors in all cases all numerical results for the stress intensity factors also exhibit domain independence the pertinent post processing routines are provided for download via the world wide web

an extended stochastic finite element method for solving stochastic partial differential equations on random domains abstract recently a new strategy was proposed to solve stochastic partial differential equations on random domains it is based on the extension to the stochastic framework of the extended finite element method x fem this method leads by a direct calculus to an explicit solution in terms of the variables describing the randomness on the geometry it relies on two major points the implicit representation of complex geometries using random level set functions and the use of a galerkin approximation at both stochastic and deterministic levels in this article we detail the basis of this technique from theoretical and technical points of view several numerical examples illustrate the efficiency of this method and compare it to other approaches

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

regularity for dirac harmonic maps into certain pseudo riemannian manifolds abstract we show the smoothness of weakly dirac harmonic maps from a closed spin riemann surface into stationary lorentzian manifolds and obtain a regularity theorem for a class of critical elliptic systems without anti symmetry structures

the analogue of picard s theorem for quasiregular mappings in dimension three the theory of quasiregular mappings has turned out to be the fight extension of the geometric parts of the theory of analytic functions in the plane to real n dimensional space the study of these mappings was initiated by re etnjak around 1966 and his main contributions to the theory is presented in the recent book 8 for the basic theory of quasiregular mappings we refer to 2 3 13 the definition is given in section 2 1 in 1967 zofi 14 raised the question of the validity o f a picard s theorem on omitted values for quasiregular mappings such a theorem appeared in 1980 in the following form

ar ω weighted inequalities for a harmonic tensors and related operators abstract we first prove the local a r weighted poincare inequality for a harmonic tensors on domain and the analogue for the projection operator then we prove the global a r weighted poincare inequality for a harmonic tensors finally using the global a r weighted inequality established for a harmonic tensors we obtain the global a r weighted poincare inequality for the projection operator

global poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation both local and global a r weighted poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation in john domains are established as applications of the poincare inequalities for green s operator the a r weighted sobolev imbedding inequalities for green s operator in john domains are also proved

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

explicit expressions for energy release rates using virtual crack extensions a new finite element technique for calculating energy release rates is presented an explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings energy release rates are calculated directly from integral expressions evaluated over singular quarter point isoparametric elements surrounding the crack tip since the energy release rates are expressed in variational form there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension the method is shown to produce very accurate solutions even with fairly coarse element meshes a similar technique for mixed mode fracture based on mutual potential energy release rates is described

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

sobolev poincare embeddings for operators on harmonic forms on manifolds we prove two weight sobolev poincare embedding theorems for compositions of the laplace beltrami operator the homotopy operator and green s operator applied to harmonic forms on manifolds these results can be used to study the lp theory of differential forms and the properties of operators

global estimates for compositions of operators applied to differential forms in this paper we obtain some global l p estimates for compositions of operators applied to differential forms satisfying the a harmonic equation these estimates indicate that the operators tdg and tg are bounded

fractional sobolev spaces and topology abstract consider the sobolev class w s p m n where m and n are compact manifolds and p 1 s 0 1 1 p we present a necessary and sufficient condition for two maps u and v in w s p m n to be continuously connected in w s p m n we also discuss the problem of connecting a map u w s p m n to a smooth map f c m n

weighted integral inequalities for conjugate a harmonic tensors abstract we first prove a local weighted integral inequality for conjugate a harmonic tensors then as an application of our local result we prove a global weighted integral inequality for conjugate a harmonic tensors in ls averaging domains which can be considered as a generalization of the classical result finally we give applications of the above results to quasiregular mappings

lp harmonic 1 forms on submanifolds in spheres abstract let m n n 3 be a complete noncompact submanifold in a sphere s n q in this paper we prove that the dimension of the space of l p harmonic 1 forms on m n must be finite for certain p 0 provided the total curvature of m n is finite this result can be regarded as a generalization of zhu fang s result on l 2 harmonic 1 forms

higher integrability of green s operator and homotopy operator abstract we prove the higher integrability and higher order imbedding theorems of the composition of homotopy operator and green s operator we also study analogues of the homotopy operator and green s operator respectively

on the strong l p hodge decomposition over complete riemannian manifolds we establish the strong lp hodge decomposition theorem and the lp poincare inequalities on differential forms over complete non compact riemannian manifolds as applications we prove some lp estimates and existence theorems for the de rham operator as well as some vanishing theorems for the lp cohomology and the lp torsion over complete non compact riemannian manifolds with suitable geometric conditions

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for elastodynamics the scaled boundary finite element method alias the consistent infinitesimal finite element cell method is developed starting from the governing equations of linear elastodynamics only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one no fundamental solution is necessary and thus no singular integrals must be evaluated general anisotropic material is analysed without any increase in computational effort boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization this method is exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions for a bounded medium symmetric static stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption a stress singularity is represented very accurately as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization

stochastic fracture mechanics by fractal finite element method this paper presents stochastic fracture mechanics analysis of linear elastic cracked structures subjected to mixed mode modes i and ii loading conditions using fractal finite element method ffem the method involves ffem for calculating fracture response characteristics statistical models of uncertainties in load material properties and crack geometry and the first order reliability method for predicting probabilistic fracture response and reliability of cracked structures the sensitivity of fracture parameters with respect to crack size required for probabilistic analysis is calculated using continuum shape sensitivity analysis numerical examples based on mode i and mixed mode problems are presented to illustrate the proposed method the results show that the predicted failure probability based on the proposed formulation of the sensitivity of fracture parameter is accurate in comparison with the monte carlo simulation results since all gradients are calculated analytically reliability analysis of cracks can be performed efficiently using ffem

on an eigenvalue problem related to the critical exponent in this paper we study the eigenvalue problem where is a smooth bounded domain and u is a positive solution of the problem such that where s is the best sobolev constant for the embedding of h10 into l2 we prove several estimates for the eigenvalues i of i i 2 n 2 and some qualitative properties of the corresponding eigenfunctions

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms