gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

positive solutions for some singular critical growth nonlinear elliptic equations let omega be a bounded domain in r n n greater than or equal to 4 with smooth boundary partial derivativeomega and the origin 0 is an element of omega mu n 2 2 2 2 2n n 2 k x is a smooth function on omega and positive somewhere we obtain existence results of positive solutions to the dirichlet problem deltau mu u x 2 k x u 2 2 u f x u on omega u 0 on partial derivativeomega for various k x and suitable number mu c 2004 elsevier ltd all rights reserved

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

topological properties of linked disclinations in anisotropic liquids a connection between the topological properties of the o 3 and so 3 sigma models based on hopf s invariant and the wess zumino term respectively and related topological concepts are indicated this is applied to the o 3 z2 and the so 3 pi sigma models where z2 and pic so 3 are point symmetry groups describing anisotropic liquids it is shown that the hopf invariant for the nematic liquid assumes integer multiples of 1 4 and similar results hold for the other sigma models applications to a topological field theory of o 3 z2 and so 3 pi sigma models are indicated

limiting behavior of the ginzburg landau functional we continue our study of the functional ee u u12 u 2 14e2 1 u 2 2dx for u h1 u r2 where u is a bounded open subset of r2 compactness results for the scaled jacobian of ue are proved under the assumption that ee ue is bounded uniformly by a function of e in addition the gamma limit of ee ue ln e 2 is shown to be e v 12 v 22 v m where v is the limit of j ue ln e j ue ue due and m is the total variation of a radon measure these results are applied to the ginzburg landau functionalfe u a hext u12 au 2 14e2 1 u 2 2 12 a hext dx with external magnetic field hext h ln e the gamma limit of fe ln e 2 is calculated to bef v a h 12 v a 22 v m a h 22 where v is as before and a is the limit of ae ln e

small cancellation theory and automatic groups part ii the main results obtained in part i can be rephrased in the language of root systems and euclidean planar tessellations as saying that finite a1 x a1 and a 2 piecewise euclidean abbreviated pe complexes of non positive curvature have automatic fundamental groups a 2 dimensional complex is a a 1 a1 respectively a2 pe complex if every 2 cell can be identified with a unit square resp equilateral triangle in the euclidean plane in such a way that the induced metrics agree on intersections such a complex has non positive curvature if the link of each vertex contains no circuits without backtracking of length less than 2 n here we prove using similar techniques that finite 2 complexes modelled on the other two root systems b 2 and g2 corresponding to tessellations by 45 right angle triangles and by 30 60 right angle triangles also have automatic fundamental groups theorems 2 3 and 3 2 to be precise

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

motion and pinning of discrete interfaces we describe the motion of interfaces in a two dimensional discrete environment by coupling the minimizing movements approach by almgren taylor and wang and a discrete to continuous analysis we show that below a critical ratio of the time and space scalings we have no motion of interfaces pinning while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by almgren and taylor the critical regime is much richer exhibiting a pinning threshold small setsmove large sets are pinned partial pinning portions of interfaces may not move pinning after an initial motion possibly to a non convex limit set quantization of the interface velocity and non uniqueness effects

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

dislocation microstructures and the effective behavior of single crystals we consider single crystal plasticity in the limiting case of infinite latent hardening which signifies that the crystal must deform in single slip at all material points this requirement introduces a nonconvex constraint and thereby induces the formation of fine scale structures we restrict attention throughout to linearized kinematics and deformation theory of plasticity which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well defined variational structure analogous to elasticity we first study a scale invariant local problem we show that by developing microstructures in the form of sequential laminates of finite depth crystals can beat the single slip constraint i e the macroscopic relaxed constitutive behavior is indistinguishable from multislip ideal plasticity in a second step we include dislocation line energies and hence a length scale into the model different regimes lead to several possible types of microstructure patterns we present constructions which achieve the various optimal scaling laws and discuss the relation with experimentally known scalings such as the hall petch law

complex hyperbolic hyperplane complements we study spaces obtained from a complete finite volume complex hyperbolic n manifold m by removing a compact totally geodesic complex n 1 submanifold s the main result is that the fundamental group of m setminus s is relatively hyperbolic relative to fundamental groups of the ends of m setminus s and m setminus s admits a complete finite volume a regular riemannian metric of negative sectional curvature it follows that for n 1 the fundamental group of m setminus s satisfies mostow type rigidity has solvable word and conjugacy problems has finite asymptotic dimension and rapid decay property satisfies borel and baum connes conjectures is co hopf and residually hyperbolic has no nontrivial subgroups with property t and has finite outer automorphism group furthermore if m is compact then the fundamental group of m setminus s is biautomatic and satisfies strong tits alternative

real hyperbolic hyperplane complements in the complex hyperbolic plane abstract this paper studies riemannian manifolds of the form m s where m 4 is a complete four dimensional riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane c h 2 and s is a compact totally geodesic codimension two submanifold whose induced riemannian metric is modeled on the real hyperbolic plane h 2 in this paper we write the metric on c h 2 in polar coordinates about s compute formulas for the components of the curvature tensor in terms of arbitrary warping functions theorem 7 1 and prove that there exist warping functions that yield a complete finite volume riemannian metric on m s whose sectional curvature is bounded above by a negative constant theorem 1 1 1 the cases of m s modeled on h n h n 2 and c h n c h n 1 were studied by belegradek in 4 and 3 respectively one may consider this work as part 3 to this sequence of papers

singular kernels multiscale decomposition of microstructure and dislocation models we consider a model for dislocations in crystals introduced by koslowski cuitino and ortiz which includes elastic interactions via a singular kernel behaving as the h1 2 norm of the slip we obtain a sharp interface limit of the model within the framework of convergence from an analytical point of view our functional is a vector valued generalization of the one studied by alberti bouchitte and seppecher to which their rearrangement argument no longer applies instead we show that the microstructure must be approximately one dimensional on most length scales and we exploit this property to derive a sharp lower bound

defects in nematic shells a γ convergence discrete to continuum approach in this paper we rigorously investigate the emergence of defects on nematic shells with a genus different from one this phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field to this end we consider a discrete xy system on the shell m described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell defects emerge when we let the mesh size of the triangulation go to zero namely in the discrete to continuum limit in this paper we investigate the discrete to continuum limit in terms of convergence in two different asymptotic regimes the first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell m via the poincare hopf theorem the second scaling produces the so called renormalized energy that governs the equilibrium of the configurations with defects

peripheral fillings of relatively hyperbolic groups in this paper a group theoretic version of dehn surgery is studied starting with an arbitrary relatively hyperbolic group g we define a peripheral filling procedure which produces quotients of g by imitating the effect of the dehn filling of a complete finite volume hyperbolic 3 manifold m on the fundamental group 1 m the main result of the paper is an algebraic counterpart of thurston s hyperbolic dehn surgery theorem we also show that peripheral subgroups of g almost have the congruence extension property and the group g is approximated in an algebraic sense by its quotients obtained by peripheral fillings

existence results for brezis nirenberg problems with hardy potential and singular coefficients abstract in this paper we consider the existence of non trivial solutions to semi linear brezis nirenberg type problems with hardy potential and singular coefficients first we shall study the corresponding eigenvalue problem and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions secondly we consider the extremal functions of the best embedding constant and get some crucial estimates for the cut off function of the extremal functions thirdly applying different variational theorems for distinct cases of those parameters appearing in the equation we obtain two existence results for non trivial solutions to semi linear brezis nirenberg type problems our existence results are divided into non resonant and resonant cases

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

a boundary value problem related to the ginzburg landau model we analyze the ginzburg landau equation for a superconductor in the case of a 2 dimensional model a cylindrical conductor with a magnetic field parallel to the axis this amounts to find the extrema of the free energy mathcal a kappa 1 2 int limits omega nabla ia phi 2 b a 2 kappa 4 phi 2 1 2 dx where is a bounded domain with smooth boundary in 2 a a1 a2 the vector potential ba 1a2 2a1 the magnetic field a complex field we describe the connected components of the maximal configuration space i e of the set of all a with components in the sobolev spaceh1 and such that 1 on the boundary modulo the action of the gauge group in the critical case 1 we give a complete description of the minimal configurations in each component

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

multiple solutions for the brezis nirenberg problem with a hardy potential and singular coefficients by energy estimates and by establishing a local ps condition we establish the multiplicity of solutions to a class of semi linear brezis nirenberg type problems with a hardy term and singular coefficients via the pseudo index theory

variational analysis of the asymptotics of the xy model in this paper we consider the xy n dimensional possibly anisotropic spin type model and by comparison with a ginzburg landau type functional we perform a variational analysis in the limit when the number of particles diverges in particular we show how the appearance of vortex like singularities can be described by properly scaling the energy of the system through a convergence procedure we also address the problem in the case of long range interactions and solve it in two dimensions

bilinearization of multidimensional topological magnets a classical magnetic model with compact su 2 and non compact su 1 1 spin phase space admitting the hirota bilinear form is presented in an arbitrary number of space dimensions the essential point of the construction is the presence of a velocity field with non trivial vorticity tensor related to a topological current density the model modifies in particular a variety of familiar topological equations such as the heisenberg ferromagnets and the 2 1 dimensional o 3 sigma model using the bilinear representation several special cases and exact solutions of physical interest spin waves domain walls and vortices are considered

rational subgroups of biautomatic groups centralizers of finite subsets in biautomatic groups are them selves biautomatic every polycyclic subgroup of a biautomatic group is abelian by finite

singularities of energy minimizing maps from the ball to the sphere examples counterexamples and bounds we study maps qp from domains q in r3 to s2 which have prescribed boundary value functions 4 and which minimize dirichlet s integral energy such p s can have isolated singular points and our principal concern is the number and arrangement of such singularities our main result is that the number of singular points is bounded above by a constant times dirichlet s integral of 4 on d q and we show that this linear law is the best possible in two different senses furthermore singularities of minimizing q s sometimes have peculiar and counterintuitive properties i we illustrate by example that the mapping area jacobian integral of 4 can be zero although qp has many singular points ii we also show how to construct uniquely minimizing qp s having many singular points stacked up vertically near d q like bubbles in a pan of water about to boil iii finally we show by example that symmetries of q and 4 do not insure corresponding symmetries of qp and its singularities in particular singularities in p can be unstable under small perturbations of 4

the line tension approximation as the dilute limit of linear elastic dislocations we prove that the classical line tension approximation for dislocations in crystals that is the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length follows as the limit of regularized linear elasticity as the lattice parameter becomes increasingly small or equivalently as the dislocation measure becomes increasingly dilute we consider two regularizations of the theory of linear elastic dislocations a core cutoff and a mollification of the dislocation measure we show that both regularizations give the same energy in the limit namely an energy defined on matrix valued divergence free measures concentrated on lines the corresponding self energy per unit length b t which depends on the local burgers vector and orientation of the dislocation does not however necessarily coincide with the self energy per unit length 0 b t obtained from the classical theory of the prelogarithmic factor of linear elastic straight dislocations indeed microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its h1 elliptic envelope

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

a variational model for dislocations in the line tension limit we study the interaction of a singularly perturbed multiwell energy with an anisotropic nonlocal regularizing term of h1 2 type and a pinning condition this functional arises in a phase field model for dislocations which was recently proposed by koslowski cuitino and ortiz but it is also of broader mathematical interest in the context of the dislocation model we identify the limit of the energy in all scaling regimes for the number n of obstacles the most interesting regime is n ln where is a nondimensional length scale related to the size of the crystal lattice in this case the limiting model is of line tension type one important feature of our model is that the set of energy wells is periodic and hence not compact thus a key ingredient in the proof is a compactness estimate up to a single translation for finite energy sequences which generalizes earlier results from alberti bouchitte and seppecher for the two well problem with a h1 2 regularization

discrete crystal elasticity and discrete dislocations in crystals this article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals the theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups differential forms and operators and a theory of integration of forms in particular we define the lattice complex of a number of commonly encountered lattices including body centered cubic and face centered cubic lattices we show that material frame indifference naturally leads to discrete notions of stress and strain in lattices lattice defects such as dislocations are introduced by means of locally lattice invariant but globally incompatible eigendeformations the geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations such as the dislocation density tensor the equation of conservation of burgers vector kroner s relation and mura s formula for the stored energy we additionally supply conditions for the existence of equilibrium displacement fields we show that linear elasticity is recovered as the limit of harmonic lattice statics as the lattice parameter becomes vanishingly small we compute the limit of dilute dislocation distributions of dislocations and we show that the theory of continuously distributed linear elastic dislocations is recovered as the limit of the stored energy as the lattice parameter and burgers vectors become vanishingly small

ginzburg landau functionals and renormalized energy a revised γ convergence approach abstract we give short and self contained proofs of convergence results for ginzburg landau energy functionals in two dimensions in the logarithmic energetic regime in particular we derive the renormalized energy by convergence

asymptotic behaviour of a pile up of infinite walls of edge dislocations we consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations the dislocations are represented by points in a plane and they are arranged in vertical walls each wall is free to move in the horizontal direction the system is described by a discrete energy depending on the one dimensional horizontal positions x i 0 of the n walls the energy contains contributions from repulsive pairwise interactions between all walls a global shear stress forcing the walls to the left and a pinned wall at x 0 that prevents the walls from leaving through the left boundary we study the behaviour of the energy as the number of walls n tends to infinity and characterise this behaviour in terms of convergence there are five different cases depending on the asymptotic behaviour of the single dimensionless parameter n corresponding to beta n ll 1 n 1 n ll beta n ll 1 and beta n gg 1 and the two critical regimes n 1 n and n 1 as a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes the results shed new light on the open problem of upscaling large numbers of dislocations we show how various existing upscaled models arise as special cases of the theorems of this paper the wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities this additional information is encoded in the limit of the dimensionless parameter n

topological properties of linked disclinations and dislocations in solid continua linked disclinations in three dimensional solid continua are studied via the wess zumino term and related topological concepts for the transformation group gl3 3 r and its quotient spaces gl 3 r pi 3 and so 3 pi 3 where pi 3 represents point symmetry groups of anisotropic solids the relation with the topological properties of anisotropic liquids is indicated dislocations are treated as dipolar pairs of disclination loops and alternatively using kroner s approach of material connections linking of dislocations is studied via the hopf invariant and gauss linking number and a connection with ashtekar s new variables is pointed out

analysis of boundary conditions for crystal defect atomistic simulations numerical simulations of crystal defects are necessarily restricted to finite computational domains supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment this work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed we formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers using this foundation we then present rigorous error estimates for i a truncation method dirichlet boundary conditions ii periodic boundary conditions iii boundary conditions from linear elasticity and iv boundary conditions from nonlinear elasticity numerical results confirm the sharpness of the analysis

closed vortex type solitons with hopf index the structure of three dimensional solitons with non trivial hopf index is investigated for the s2 nonlinear sigma model it is shown that the corresponding regular solutions are of closed vortex type the author prove the existence of regular vortex like solutions which are used for the approximation of solitons with large values of hopf index

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

existence and stability of a screw dislocation under anti plane deformation we formulate a variational model for a geometrically necessary screw dislocation in an anti plane lattice model at zero temperature invariance of the energy functional under lattice symmetries renders the problem non coercive nevertheless by establishing coercivity with respect to the elastic strain and a concentration compactness principle we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model

γ convergence analysis of systems of edge dislocations the self energy regime this paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity the dislocations are introduced as point topological defects of the displacement gradient fields following the core radius approach we introduce a parameter varepsilon 0 representing the lattice spacing of the crystal we remove a disc of radius varepsilon around each dislocation and compute the elastic energy stored outside the union of such discs namely outside the core region then we analyze the asymptotic behaviour of the elastic energy as varepsilon rightarrow 0 in terms of convergence we focus on the self energy regime of order log frac 1 varepsilon we show that configurations with logarithmic diverging energy converge up to a subsequence to a finite number of multiple dislocations and we compute the corresponding limit

convergence and non convergence of many particle evolutions with multiple signs we address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity we consider two types of particles called positive and negative same sign particles repel each other and opposite sign particles attract each other the interaction potential is the same for all particles up to the sign and has a logarithmic singularity at zero the central example of such systems is that of dislocations in crystals because of the singularity in the interaction potential the discrete evolution leads to blow up in finite time we remedy this situation by regularising the interaction potential at a length scale delta n 0 which converges to zero as the number of particles n tends to infinity we establish two main results the first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled pdes which describe the evolution of their continuum densities in the setting of dislocations these pdes are known as the groma balogh equations in the proof we rely on both the theory of lambda convex gradient flows to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta n regularised version of the groma balogh equations and a priori estimates for the groma balogh equations to pass to the small regularisation limit in a functional setting based on orlicz spaces in order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta n to converge to zero sufficiently slowly the second result is a counterexample demonstrating that if delta n converges to zero sufficiently fast then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the groma balogh equations these results show how the validity of the groma balogh equations as the limit of many particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised

dynamics for systems of screw dislocations the goal of this paper is the analytical validation of a model of cermelli and gurtin arch ration mech anal 148 1999 pp 3 52 for an evolution law for systems of screw dislocations under the assumption of antiplane shear the motion of the dislocations is restricted to a discrete set of glide directions which are properties of the material the evolution law is given by a maximal dissipation criterion leading to a system of differential inclusions short time existence uniqueness cross slip and fine cross slip of solutions are proved

topological properties of static and dynamic defect configurations in ordered liquids abstract the topology of linked disclinations is studied in uniaxial nematic liquids and in anisotropic liquids with an order parameter space so 3 pi 3 in these models pi 3 are the finite point symmetry groups of 3 space with applications to helium 3 pi 3 i biaxial nematic liquid pi 3 d2 and anisotropic super cooled liquids pi 3 groups of platonic solids the topological properties are studied via hopf s invariant of the o 3 model and its relation to the wess zumino term of the so 3 pi 3 model in an orthonormal drei bein representation of so 3 dynamic processes are topology changing during intersection of disclinations and are studied via magnetic n pole singularities and the instanton number in an electromagnetic formalism the connection with tunneling amplitudes in a so 3 yang mills theory is indicated applications of the theory to topological fluid dynamics is worked out for the uniaxial nematic liquid and indicated for the so 3 spin liquid

metastability and dynamics of discrete topological singularities in two dimensions a γ convergence approach this paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions based on convergence we consider discrete systems described by scalar functions defined on a square lattice and governed by periodic interaction potentials our main motivation comes from xy spin systems described by the phase parameter and screw dislocations described by the displacement function for these systems we introduce a discrete notion of vorticity as the lattice spacing tends to zero we derive the first order limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices as a byproduct of this analysis we show that such systems exhibit increasingly many metastable configurations of singularities therefore we propose a variational approach to the depinning and dynamics of discrete vortices based on minimizing movements we show that letting first the lattice spacing and then the time step of the minimizing movements tend to zero the vortices move according with the gradient flow of the renormalized energy as in the continuous ginzburg landau framework

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we