equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

partial regularity of the landau de gennes energy model for liquid crystals abstract we discuss the partial regularity of non trivial minimizers for the landau de gennes energy functional in a bounded domain of r 3 off some relatively closed subset z whose 1 dimensional hausdorff measure is zero we prove that the minimizers are analytic

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

a general comparison theorem for p harmonic maps in homotopy class abstract we prove a general comparison result for homotopic finite p energy c 1 p harmonic maps u v m n between riemannian manifolds assuming that m is p parabolic and n is complete and nonpositively curved in particular we construct a homotopy through constant p energy maps which turn out to be p harmonic when n is compact moreover we obtain uniqueness in the case of negatively curved n this generalizes a well known result in the harmonic setting due to r schoen and s t yau

a quantization property for static ginzburg landau vortices for any critical point of the complex ginzburg landau functional in dimension 3 we prove that for large coupling constants if the energy of this critical point on a ball of a given radius r is relatively small compared to r log then the ball of half radius contains no vortex the modulus of the solution is larger than we then show how this property can be applied to describe limiting vortices as e 0 2001 john wiley sons inc

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

homogenization of high frequency wave propagation in linearly elastic layered media using a continuum irving kirkwood theory abstract this article presents an application of a recently developed continuum homogenization theory inspired by the classical work of irving and kirkwood to the homogenization of plane waves in layered linearly elastic media the theory explicitly accounts for the effects of microscale dynamics on the macroscopic definition of stress it is shown that for problems involving high frequency wave propagation the macroscopic stress predicted by the theory differs significantly from classical homogenized stress definitions the homogenization of plane waves is studied to illustrate key aspects and implications of the theory including the characteristics of the homogenized macroscopic stress and the influence of frequency on the determination of an intermediate asymptotic length scale in addition a method is proposed for predicting the homogenized stress field in a one dimensional bar subjected to a frequency dependent forced vibration using only knowledge of the boundary conditions and the material s dispersion solution furthermore it is shown that due to the linearity of the material the proposed method accurately predicts the homogenized stress for any time varying displacement or stress boundary condition that can be expressed as a sum of time periodic signals

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

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

liquid crystals and the monotonicity formula abstract we give numerical evidence that the monotonicity formula holds for the liquid crystal functional and study part of the associated bifurcation diagram

numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces this article studies the numerical approximation of harmonic maps mto surfaces i e critical pomts for the dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold an iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations numerical simulations illustrate the performance of the algorithm for curved domains

brownian motion and the formation of singularities in the heat flow for harmonic maps we develop a general framework for a stochastic interpretation of certain nonlinear pdes on manifolds the linear operation of takin expectations is replaced by the concept of martingale means namely the notion of deterministic starting points of martingales with respect to the levi civita connection ending up at a prescribed state we formulate a monotonicity condition for the riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation such estimates lead to simple criteria for blow ups in the nonlinear heat flow for harmonic maps with small initial energy

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

a homogenization method for thermomechanical continua using extensive physical quantities this article proposes a continuum thermomechanical homogenization method inspired by the irving kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics this method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities the resulting equation for macroscopic stress affords a rational comparison with the widely used hill mandel stress deformation condition while the one for heat flux reduces under certain assumptions to a hill mandel like condition involving heat flux and the gradient of temperature

novel formulations of microscopic boundary value problems in continuous multiscale finite element methods abstract this article explores the use of a homogenization method inspired by the classical irving kirkwood procedure in the continuous multiscale modeling of elastic solids within the context of the finite element method this homogenization method gives rise to a broad range of allowable boundary conditions for the rve which in turn yield a rich spectrum of estimates of the macroscopic stress response for two representative problems involving heterogeneous elastic bodies

stability and asymptotics in nematic liquid crystals under a small dirichlet data and a non constant magnetic field abstract we consider the stability of a critical point and asymptotics of the minimizers of the free energy functional with a small dirichlet boundary data under a non constant exterior magnetic field in nematic liquid crystals in the author s previous papers we studied the stability of a critical point under the more general hypotheses but we had to assume the curl free condition on the critical point for technical reason however in this paper without the curl free condition we can get the similar result in the special case where the elastic coefficients are equal

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

on a common generalization of koszul duality and tilting equivalence abstract we propose a new definition of koszulity for graded algebras where the degree zero part has finite global dimension but is not necessarily semi simple the standard koszul duality theorems hold in this setting

weak solutions to the heat flow for surfaces of prescribed mean curvature in this talk we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature i e the existence for the cauchy dirichlet problem to parabolic systems of the type tu u 2 h u d1u d2u in b 0 u uo on par b 0 where h r3 r is a bounded continuous function satisfying an isoperimetric condition b the unit ball in r2 and u b 0 r3 as one of the possible applications we show that the problem has a solution with values inbr r3 whenever uo b br and furthermore there holds br h 3 2r h d 9 2 h a 1 r for a br the results that will be presented in the talk are joint work with verena bogelein und chrsitoph scheven from erlangen frank duzaar department of mathematics university erlangen nurnberg cauerstr 11 91056 erlangen germany e mail address frank duzaar zuv uni erlangen de date december 29 2011 1

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

an elementary partial regularity proof for vector valued obstacle problems soit r n et m r n des ensembles ouverts bornes on suppose n 3 n 2 et on requiert que la frontiere de m est une hyper surface lisse de classe c 2 l espace de fonction h 1 2 m des fonctions de comparaison consiste en toutes les fonctions de sobolev v h 1 2 r n avec la propriete v x m pour presque tous les points x dans si u h 1 2 m minimise localement l energie alors il y a un sous ensemble ouvert 0 de tel que u c 1 0 r n pour tout 0 1 h n 2 0 0 et x 0 si et seulement si lim 1 0 r 2 n br x du 2 dz 0

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

quaternionic maps between hyperkahler manifolds quaternionic maps q maps between hyperkahler manifolds are quaternionic analogue of cauchy riemann equations between kahler manifolds we provide a necessary and sufficient condition on when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkahler 2 spheres and give examples of q maps which cannot be holomorphic when the domain is real 4 dimensional we analyze the structure of the blow up set of a sequence of q maps and show that the singular set of a stationary q map is h1 rectifiable

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

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

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

quasi hereditary algebras and generalized koszul duality abstract we present an easily applicable sufficient condition for standard koszul algebras to be koszul with respect to if a quasi hereditary algebra is koszul with respect to then and the yoneda extension algebra of are koszul dual in a sense explained below implying in particular that their bounded derived categories of finitely generated graded modules are equivalent we also prove that the extension algebra of is koszul in the classical sense

the singular set of stationary solution for semilinear elliptic equations with supercritical growth abstract in this paper we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations for almost all supercritical exponents we improve the upper bounds of the hausdorff dimension of the singular set which are derived by pacard to the optimal ones

multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method abstract this paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with voronoi cell vcfem for microstructural modeling the voronoi cell finite element model originates from dirichlet tessellation of a representative material element or a base cell in the microstructure homogenized material coefficients for a global displacement finite element model are generated by vcfem analysis using periodic boundary conditions on the base cell following the macroscopic analysis the local vcfem analysis is implemented to depict the true evolution of microstructural stresses and strains various numerical examples are executed for validating the effectiveness of vcfem macro micro modeling of elastic materials the effect of size shape orientation and distribution of heterogeneities on the local and global response are examined

neohookean deformations of annuli existence uniqueness and radial symmetry let mathbb x x in mathbb r 2 r x r and mathbb y y in mathbb r 2 r ast y r ast be annuli in the plane we consider the class fancyscript f mathbb x mathbb y of all orientation preserving homeomorphisms h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev space fancyscript w 1 2 mathbb x mathbb y which keep the boundary circles in the same order this means that lim x searrow r h x r ast and lim x nearrow r h x r ast we study the neohookean energy integral mathcal e h int limits mathbb x dh 2 phi rm det dh quad rm for h in fancyscript f mathbb x mathbb y quad quad quad 1 where phi in fancyscript c infty 0 infty is positive and strictly convex we assume in addition that the function psi z 1 ddot phi z and its derivative extend continuously to 0 with 0 0 then we prove

mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading we present a generalization of the classical mathematical homogenization theory aimed at accounting for finite unit cell distortions which gives rise to a nonperiodic asymptotic expansion we introduce an auxiliary macro deformed configuration where the overall cauchy stress is defined and nonperiodic boundary conditions verification studies against a direct numerical simulation demonstrate the versatility of the proposed method

harmonic maps from noncompact riemannian manifolds with non negative ricci curvature outside a compact set in this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete noncompact riemannian manifold m g with sobolev constant s2 m 0 and ricci curvature ric m 0 outside some compact subset into a complete manifold of nonpositive curvature or a regular ball in particular we prove the uniqueness and existence of bounded harmonic functions on m g

phase transition for potentials of high dimensional wells for a potential function f w r k rc that attains its global minimum value at two disjoint compact connected submanifolds n in r k we discuss the asymptotics as 0 of minimizers u of the singular perturbed functional e u d r jruj 2 c 1 2 f u dx under suitable dirichlet boundary data g w r k in the expansion of e u with respect to 1 we identify the first order term by the area of the sharp interface between the two phases an area minimizing hypersurfacea and the energyc f of minimal connecting orbits between n c and n and the zerothorder term by the energy of minimizing harmonic maps into n both under the dirichlet boundary condition on and a very interesting partially constrained boundary condition on the sharp interfacea 2011 wiley periodicals inc

the uniqueness of tangent cones for yang mills connections with isolated singularities we proved a uniqueness theorem of tangent connections for a yang mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity we also obtained control over the rate of the asymptotic convergence of the connection to the tangent connection if furthermore the connection is stationary or the tangent connection is integrable with a stronger result in the latter case there are parallel results for the cones at infinity of a yang mills connection on an asymptotically flat manifold we also gave an application of our methods to the yang mills flow and proved that the yang mills flow exists for all time and has asymptotic limit if the initial value is close to a smooth local minimizer of the yang mills functional

some regularity results for p harmonic mappings between riemannian manifolds abstract let m be a c 2 smooth riemannian manifold with boundary and n a complete c 2 smooth riemannian manifold we show that each stationary p harmonic mapping u m n whose image lies in a compact subset of n is locally c 1 for some 0 1 provided that n is simply connected and has non positive sectional curvature we also prove similar results for minimizing p harmonic mappings with image being contained in a regular geodesic ball moreover when m has non negative ricci curvature and n is simply connected with non positive sectional curvature we deduce a gradient estimate for c 1 smooth weakly p harmonic mappings from which follows a liouville type theorem in the same setting

on global singularities of sobolev mappings we define various invariants for sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong sobolev topology we show that these invariants classify various types of global singularities of sobolev maps these invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the sobolev space

rigidity of teichmuller space we prove the holomorphic rigidity conjecture of teichmuller space which loosely speaking states that the action of the mapping class group uniquely determines the teichmuller space as a complex manifold the method of proof is through harmonic maps we prove that the singular set of a harmonic map from a smooth n dimensional riemannian domain to the weil petersson completion overline mathcal t of teichmuller space has hausdorff dimension at most n 2 and moreover u has certain decay near the singular set combining this with the earlier work of schumacher siu and jost yau we provide a proof of the holomorphic rigidity of teichmuller space in addition our results provide as a byproduct a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by farb masur and yeung

the weyl extension algebra of gl2 f p abstract we compute the yoneda extension algebra of the collection of weyl modules for g l 2 over an algebraically closed field of characteristic p 0 by developing a theory of generalised koszul duality for certain 2 functors one of which controls the rational representation theory of g l 2 over such a field

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

biaxial escape in nematics at low temperature abstract in the present work we study minimizers of the landau de gennes free energy in a bounded domain r 3 we prove that at low temperature minimizers do not vanish even for topologically non trivial boundary conditions this is in contrast with a simplified ginzburg landau model for superconductivity studied by bethuel brezis and helein merging this with an observation of canevari we obtain as a corollary the occurrence of biaxial escape the tensorial order parameter must become strongly biaxial at some point in in particular while it is known that minimizers cannot be purely uniaxial we prove the much stronger and physically relevant fact that they lie in a different homotopy class

compactness theorems for coupled yang mills fields in this paper we consider coupled yang mills fields on vector bundle e over compact riemannian manifold m under appropriate conditions on the curvature and the higgs field two compactness theorems are proved

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

monotonicity formulas for extrinsic triharmonic maps and the triharmonic lane emden equation abstract we derive monotonicity formulas and e regularity results for extrinsic triharmonic maps and triharmonic lane emden equations as an application we prove partial regularity results for both types of equations in the supercritical regime

prediction of the mechanical behavior of nonlinear heterogeneous systems by multi level finite element modeling an accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented this method is based on the classical homogenization theory assuming local spatial periodicity of the microstructure consequently the microstructure is identified by a representative volume element rve with conformity of opposite boundaries at any stage of the deformation process the local macroscopic stress is obtained by applying the local macroscopic deformation represented by the deformation tensor on a unique rve through imposing appropriate boundary conditions and averaging the resulting rve stress field if the assumption of local periodicity of the morphology is valid this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation the homogenization method was implemented in a multi level finite element program with meshes on macroscopic level mesh of entire structure and microscopic level meshes of rves the performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet

singular points of harmonic maps from 4 dimensional domains into 3 spheres in this article we show that if a stable stationary harmonic map u from a domain in r4 into s3 has an isolated singular point then the mapping degree deg u of u at is 1 or 1 furthermore the complete characterization of stable stationary tangent maps from b4 to s3 is given