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

local minimizers of the ginzburg landau energy with magnetic field in three dimensions we establish the existence of locally minimizing vortex solutions to the full ginzburg landau energy in three dimensional simply connected domains with or without the presence of an applied magnetic field the approach is based upon the theory of weak jacobians and applies to nonconvex sample geometries for which there exists a configuration of locally shortest line segments with endpoints on the boundary

on the derivation of the numerical permittivity and stability of the cds fdtd implementation of high order constitutive relations recently the derivation of the numerical permittivity and stability of the finite difference time domain fdtd implementation of a 4 pole complex rational function by using the central difference scheme cds have been studied in this communication the accuracy and the stability limitation of this scheme are investigated in addition alternative fdtd implementation based on the mobius transformation mt is presented it is observed that the cds implementation is less accurate and more stability restrictive than the mt counterpart

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

fdtd dispersive modeling of human tissues based on quadratic complex rational function we propose a dispersive finite difference time domain fdtd suitable for the electromagnetic analysis of human tissues the dispersion relation of biological tissues is characterized by a quadratic complex rational function qcrf that leads to an accurate fdtd algorithm in 400 mhz 3 ghz qcrf coefficients are extracted by applying the complex curve fitting technique without initial guess numerical examples are used to illustrate the computational accuracy and stability of qcrf based fdtd

inverse spectral transform analysis of a nonlinear schrodinger equation with x dependent coefficients abstract using an extended akns zs formalism developed in earlier work we present a systematic inverse spectral transform analysis and solution of the generalized nonlinear schrodinger equation i q t fq xx 2fq q 2 2q x f x q 2 d x 0 which arises in certain physical problems involving inhomogeneous interactions or media physically interesting forms of the inhomogeneity function f x are discussed that support localized solutions and permit non dissipative tunneling of lump like solutions through the inhomogeneity

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

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

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

stable configurations in superconductivity uniqueness multiplicity and vortex nucleation we find new stable solutions of the ginzburg landau equation for high superconductors with exterior magnetic field h ex first we prove the uniqueness of the meissner type solution then we prove in the case of a disc domain the coexistence of branches of solutions with n vortices of degree one for any n not too high and for a certain range of h ex and describe these branches finally we give an estimate on the nucleation energy barrier to pass continuously from a vortexless configuration to a configuration with a centered vortex

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

lorentz space estimates for the ginzburg landau energy abstract in this paper we prove novel lower bounds for the ginzburg landau energy with or without magnetic field these bounds rely on an improvement of the vortex balls construction estimates by extracting a new positive term in the energy lower bounds this extra term can be conveniently estimated through a lorentz space norm on which it thus provides an upper bound the lorentz space l 2 we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results

surface embeddability approach to the dynamics of the inhomogeneous heisenberg spin chain the surface embeddability approach of lund and regge is applied to the classical inhomogeneous heisenberg spin chain to study the class of inhomogeneity functions f for which the spin evolution equation and its gauge equivalent generalized nonlinear schrodinger equation gnlse are exactly solvable writing the spin vector s x t as xr and identifying r x t with a position vector generating a surface we show that the kinematic equation satisfied by r implies certain constraints on the admissible geometries of this surface these constraints together with the gauss mainardi codazzi equations enable us to express the coefficient of the second fundamental form as well as f in terms of the metric coefficients g and its derivatives for arbitrary time independent g explicit solutions for the gnlse can also be found in terms of the same quantities of the admissible surfaces generated by r a special class that emerges naturally is that of surfaces of revolution explicit solutions for r and s are found

on the transition to the normal phase for superconductors surrounded by normal conductors abstract for a cylindrical superconductor surrounded by a normal material we discuss transition to the normal phase of stable locally stable and critical configurations associated with those phase transitions we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide in particular when the conductivity ratio of the superconducting and the normal material is large we show that the aforementioned critical magnetic fields coincide thereby proving that the transition to the normal phase is sharp one key ingredient in the paper is the analysis of an elliptic boundary value problem involving transmission boundary conditions another key ingredient involves a monotonicity result with respect to the magnetic field strength of the first eigenvalue of a magnetic schrodinger operator with discontinuous coefficients

stability considerations for the direct fdtd implementation of the dispersive quadratic complex rational function models in this communication detailed stability considerations for the direct finite difference time domain implementation of the dispersive quadratic complex rational function models are investigated it is shown that the time step stability limit of this numerical scheme is more restrictive than the previously published constrain in addition it is found that by using high spatial resolution the stability limit tends to retain the classical courant friedrichs lewy constrain numerical examples are included to verify these results

homotopy classification of minimizers of the ginzburg landau energy and the existence of permanent currents our objective is to explain the phenomenon of permanent currents within the context of the ginzburg landau model for superconductors using variational techniques we make a connection between the formation of permanent currents and the topology of the superconducting sample

topology and sobolev spaces abstract consider the sobolev class w 1 p m n where m and n are compact manifolds we present some sufficient conditions which guarantee that w 1 p m n is path connected we also discuss cases where w 1 p m n admits more than one component there are still a number of open problems especially concerning the values of p where a change in homotopy classes occurs

pinning with a variable magnetic field of the two dimensional ginzburg landau model abstract we study the ginzburg landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain supposing that the ginzburg landau parameter and the intensity of the magnetic field are large and of the same order we determine an accurate asymptotic formula for the minimizing energy this asymptotic formula displays the influence of the pinning term also we discuss the existence of non trivial solutions and prove some asymptotics of the third critical field

vortices and pinning effects for the ginzburg landau model in multiply connected domains we consider the two dimensional ginzburg landau model with magnetic fi eld for a superconductor with a multiply connected cross section we study energy minimizers in the london limit as the ginzburg landau parameter 1 to determine the number and asymptotic location of vortices we show that the holes act as pinning sites acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critica l value hex o ln at the critical level the pinning effect breaks down and vortices appear in the interior of the superconductor at locations that we id entify explicitly in terms of the solutions of an elliptic boundary value problem the method involves sharp upper and lower energy estimates and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes c 2005 wiley periodicals inc

metamorphoses sudden jumps in basin boundaries in some invertible maps of the plane that depend on a parameter boundaries of basins of attraction are extremely sensitive to small changes in the parameter a basin boundary can jump suddenly and as it does change from being smooth to fractal such changes are calledbasin boundary metamorphoses we prove under certain non degeneracy assumptions that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency

local minimizers with vortices to the ginzburg landau system in three dimensions we construct local minimizers to the ginzburg landau energy in certain three dimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak jacobians whenever there exists a collection of locally minimal line segments spanning the domain we can find local minimizers with arbitrarily assigned degrees with respect to each segment 2003 wiley periodicals inc

analysis of nematic liquid crystals with disclination lines we investigate the structure of nematic liquid crystal thin films described by the landau de gennes tensor valued order parameter model with dirichlet boundary conditions on the sides of nonzero degree we prove that as the elasticity constant goes to zero in the energy a limiting uniaxial nematic texture forms with a finite number of defects all of degree frac 1 2 or all of degree frac 1 2 corresponding to vertical disclination lines at those locations we also state a result on the limiting behavior of minimizers of the chern simons higgs model without magnetic field that follows from a similar proof

metastable currents in loops with josephson junctions the problem of persistent currents in loops interrupted by josephson junctions is considered we prove that persistent currents are related to local minimizers of the ginzburg landau energy functional although the order parameter is discontinuous at the junction we show that the local minimizers are related to the homotopy types of the domain therefore persistent currents will occur in any multiply connected domain if the junction strength is weak enough

on approximation of ginzburg landau minimizers by s1 valued maps in domains with vanishingly small holes abstract we consider a two dimensional ginzburg landau problem on an arbitrary domain with a finite number of vanishingly small circular holes a special choice of scaling relation between the material and geometric parameters ginzburg landau parameter vs hole radius is motivated by a recently discovered phenomenon of vortex phase separation in superconducting composites we show that for each hole the degrees of minimizers of the ginzburg landau problems in the classes of s 1 valued and c valued maps respectively are the same the presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques

radial solutions for the ginzburg landau equation with applied magnetic field abstract the purpose of this work is a systematic study of symmetric vortices for the ginzburg landau model of superconductivity along a cylinder with applied magnetic field parallel to its axis the ginzburg landau constant of the material and the degree d of the vortex are fixed for any given parameters r the radius of the cylinder and h the intensity of the applied magnetic field one can find a symmetric vortex a which satisfies the boundary conditions b r e i d and curl a b r he 3 it is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex as the boundary conditions depend smoothly on those parameters one can distinguish two main connected domains of vortices the first one defined by the boundary conditions b r e i d and ra r b r is a zone of stability where remains increasing the second one defined by the boundary conditions b r e i d and ra r b r d is a zone where some instability can occur attention is focused on the smooth branch of vortices which separates those two domains it is indexed by the parameter running from some l 0 to and the limit as decreases to l corresponds to a berger and chen type vortex

local minimizers and quasiconvexity the impact of topology the aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals open image in new window open image in new window from a purely topological point of view the basic assumptions on open image in new window are sequential lower semicontinuity with respect to w1 p weak convergence and w1 p weak coercivity and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds open image in new window and open image in new window

ginzburg landau minimizers with prescribed degrees capacity of the domain and emergence of vortices let omega subset mathbb r 2 be a simply connected domain let omega be a simply connected subdomain of omega and set a omega setminus overline omega let j be the class of complex valued maps on the annular domain a with degree 1 both on partial omega and on partial omega we consider the variational problem of minimizing the ginzburg landau energy e lambda among all maps in j because only the degrees of the map are prescribed on the boundary the set j is not closed under weak h 1 convergence we show that the attainability of the minimum of e lambda over j is determined by the value of cap a where cap a is the h 1 capacity of the domain a in contrast it is known that the existence of minimizers of e lambda among the maps with a prescribed dirichlet boundary data does not depend on this geometric characteristic when cap a ge pi such an a is denoted as either subcritical if cap a pi or critical if cap a pi we show that the global minimizers of e lambda exist for each lambda 0 and that they are vortexless when lambda is large assuming that lambda to infty we demonstrate that the minimizers of e lambda converge in h 1 a and even better to an mathbb s 1 valued harmonic map which we explicitly identify when cap a lambda 0 or ii the global minimizers exist for each lambda 0 we conjecture that the second case never occurs further for large lambda we establish that the minimizing sequences minimizers in supercritical domains develop exactly two vortices a vortex of degree 1 near partial omega and a vortex of degree 1 near partial omega

pinning effects and their breakdown for a ginzburg landau model with normal inclusions we study a ginzburg landau model for an inhomogeneous superconductor in the singular limit as the ginzburg landau parameter 1 the inhomogeneity is represented by a potential term v 14 a x 2 2 with a given smooth function a x which is assumed to become negative in finitely many smooth subdomains the normally included regions for bounded applied fields independent of the ginzburg landau parameter 1 we show that the normal regions act as giant vortices acquiring large vorticity for large fixed applied field hex for hex o ln we show that this pinning effect eventually breaks down and free vortices begin to appear in the superconducting region where a x 0 at a point set which is determined by solving an elliptic boundary value problem the associated operators are strictly but not uniformly elliptic leading to some regularity questions to be resolved near the boundaries of the normal regions

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

painleve analysis of the damped driven nonlinear schrodinger equation on applique les tests de painleve a l equation iu t u xx 2 u 2 u a x t u b x t ou a et b sont des fonctions analytiques de x et t pour etudier sous quelles conditions l equation peut etre completement integrable

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