some results on an n ginzburg landau type minimizer this paper is concerned with the asymptotic analysis of a minimizer of an n ginzburg landau type functional when the dimension n 2 the asymptotic properties were well studied such as the convergence of the minimum of the energy the behavior of the minimizer near its zero points and the quantization effects for the euler lagrange system in r 2 the author investigates those properties when the dimension n is not less than 3

symmetric vortices for two component ginzburg landau systems abstract we study ginzburg landau equations for a complex vector order parameter c 2 we consider symmetric vortex solutions in the plane r 2 x f r e i n with given degrees n z and prove the existence uniqueness and asymptotic behavior of solutions as r we also consider the monotonicity properties of solutions and exhibit parameter ranges in which both vortex profiles f f are monotone as well as parameter regimes where one component is non monotone the qualitative results are obtained by means of a sub and super solution construction and a comparison theorem for elliptic systems

stabilization of networked control systems with both network induced delay and packet dropout this paper is concerned with the mean square stabilization problem for discrete time networked control systems ncss it is assumed that control signal is sent to plant over a lossy communication channel where network induced delay and packet dropout occur simultaneously a necessary and sufficient stabilizing condition is developed in terms of the unique positive definite solutions to some coupled algebraic riccati equations cares the contributions of this paper are twofold first an existence theorem of the maximum packet dropout rate is proposed second for one dimensional single input system and the decoupled multi input system it is shown that the ncs is stabilizable iff the network induced delay and the packet dropout rate satisfy some simple algebraic inequalities if the network induced delay is known a priori the maximum packet dropout rate is given explicitly in terms of network induced delay and unstable eigenvalues of the system matrix if the packet dropout rate is known a priori the maximum allowable delay bound is also given explicitly

nonconvex variational problems with anisotropic perturbations we obtain the l 1 limit for a class of variational problems which arise as an anisotropic singular perturbation of a nonconvex functional any limit of minimizers of the perturbed problems must minimize the new variational problem which arises in the limit and the primary goal is to understand the dependence of this new variational problem on the type of perturbation chosen

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

variational reduction for ginzburg landau vortices let be a bounded domain with smooth boundary in r2 we construct non constant solutions to the complex valued ginzburg landau equation e2 u 1 u 2 u 0 in as e 0 both under zero neumann and dirichlet boundary conditions we reduce the problem of finding solutions having isolated zeros vortices with degrees 1 to that of finding critical points of a small c1 perturbation of the associated renormalized energy this reduction yields general existence results for vortex solutions in particular for the neumann problem we find that if is not simply connected then for any k 1 a solution with exactly k vortices of degree one exists

limiting properties of the second order ginzburg landau minimizers the author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0 and this paper is also concerned with the location of the zeros and the blow up points of the gradient of the minimizers of this functional finally the strong convergence of the gradient of the radial minimizers is obtained

anisotropic singular perturbations the vectorial case we obtain the l 1 limit of the sequence where e e is the family of anisotropic perturbations of the nonconvex functional of vector valued functions the proof relies on the blow up argument introduced by fonseca and muller

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

linear quadratic regulation for linear time varying systems with multiple input delays this paper studies the classic linear quadratic regulation lqr problem for both continuous time and discrete time systems with multiple input delays for discrete time systems the lqr problem for systems with single input delay has been studied in existing literature whereas a solution to the multiple input delay case is not known to our knowledge for continuous time systems with multiple input delays the lqr problem has been tackled via an infinite dimensional system theory approach and a frequency time domain approach the objective of the present paper is to give an explicit solution to the lqr problem via a simple and intuitive approach the main contributions of the paper include a fundamental result of duality between the lqr problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system the duality allows us to obtain a solution to the lqr problem via standard projection in linear space the lqr controller is simply constructed by the solution of one backward riccati difference for the discrete time case or differential for the continuous time case equation of the same order as the plant ignoring the delays

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

some remarks concerning symmetry breaking for the ginzburg landau equation abstract the correlation term introduced in 13 to describe the interaction between very far apart vortices governs symmetry breaking for the ginzburg landau equation in r 2 or bounded domains it is a homogeneous function of degree 2 and then for 2 n symmetric vortex configurations can be expressed in terms of the so called correlation coefficient ovchinnikov and sigal 13 have computed it in few cases and conjectured its value to be an integer multiple of 4 we will disprove this conjecture by showing that the correlation coefficient always vanishes and will discuss some of its consequences

a sharp interface limit for a two well problem in geometrically linear elasticity in the theory of solid solid phase transitions the deformation open image in new window of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation we study

on the chern simons higgs vortex equation without gauge fields abstract this paper is concerned with the nonself dual chern simons higgs model on r 2 with vanishing gauge fields we prove the existence of radial solutions with the topological boundary condition and the nonexistence of radial solutions with the nontopological boundary condition we also establish the asymptotic properties of solutions and derive the quantization of the potential energy

linear quadratic regulation for discrete time systems with input delay and colored multiplicative noise abstract this paper is concerned with the linear quadratic regulation lqr problem for discrete time systems involving input delay and colored multiplicative noise due to the correlation of the adjoining state this problem will be more complicated than the case of white noise in this paper the necessary and sufficient condition for the solvability of optimal control problem is presented by solving the forward and backward stochastic difference equations fbsdes moreover the optimal lqr controller is given in terms of the coupled difference equations developed in this paper

the correlation coefficient for vortices in the plane abstract for point vortices in the plane we consider the correlation coefficient of ovchinnikov and sigal generalising a result by esposito we show that it vanishes for all vortex equilibria

weak anchoring for a two dimensional liquid crystal abstract we study the weak anchoring condition for nematic liquid crystals in the context of the landau de gennes model we restrict our attention to two dimensional samples and to nematic director fields lying in the plane for which the landau de gennes energy reduces to the ginzburg landau functional and the weak anchoring condition is realized via a penalized boundary term in the energy we study the singular limit as the length scale parameter e 0 assuming the weak anchoring parameter e at a prescribed rate we also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation for e k e with 0 1 we show that defects lie on the weak anchoring boundary for 0 1 2 or for 1 2 and k small but they occur inside the bulk domain for 1 2 or 1 2 with k large

hamiltonian identities for elliptic partial differential equations abstract new identities for elliptic partial differential equations are obtained several applications are discussed in particular young s law for the contact angles in triple junction formation is proven rigorously structure of level curves of saddle solutions to allen cahn equation are also carefully analyzed

the computational modelling of branching fine structures in constrained crystals we introduce a finite element method which is piecewise continuous on the microscopic scale of the spatial resolution h but discontinuous on the mesoscopic scale h 0 1 the method is designed to capture the morphology of needle twin structures frequently found in ferric and pseudo elastic crystals namely in uniaxial ferromagnets and au milieu of the austenite martensite interfaces the approach is based on a domain decomposition method that interpolates between the scale on order of the size of crystal and the microscopic scale of finite element approximation h the scale interpolation is enabled by incorporating frequency adaptivity the visualization and analysis of the computational results presented disclose microstructures corresponding to complex scaling laws we document that the fine structures obtained by the presented method are not visible using classical formulation of the underlying variational problem and using conforming approximation of admissible sets the proposed method is suitable for non smooth relaxation and optimization when the minimizers lack the often required c1 regularity and when they exhibit fractal behavior

eigenmode solution of 2 d and 3 d electromagnetic cavities containing absorbing materials using the jacobi davidson algorithm eigenmodes of electromagnetic cavities containing absorbing dielectric materials are determined using an adaptation of the jacobi davidson technique to solve discrete matrix eigenequations derived from maxwell s equations the discretisations obtained using finite difference and finite integration methods give rise to non hermitian matrices having complex eigenvalues and the jacobi davidson method is shown to be applicable even when very low q cavity eigenmodes exist in the presence of highly lossy dielectric or permeable materials examples are given of eigensolutions for both 2 d cylindrically symmetric and 3 d electromagnetic operators

a compactness result in the gradient theory of phase transitions in the plane as 0 this functional favours jraj 1 and penalizes singularities where jrraj concentrates our main result is a compactness theorem if fe a g 0 is uniformly bounded then fra g 0 is compact in l thus in the limit 0 a solves the eikonal equation jraj 1 almost everywhere our analysis uses entropy relations and the div curl lemma adopting tartar s approach to the interaction of linear di erential equations and nonlinear algebraic relations

two results on entire solutions of ginzburg landau system in higher dimensions in this short article we prove two results on the ginzburg landau system of equations u u u 2 1 where u c2 rn rm n m 1 first we prove a liouville type theorem which asserts that every solution u satisfying rn u 2 1 2 is constant and of unit norm provided n 4 here m 1 in our second result we give an answer to a question raised by brezis open problem 3 of proceedings of the symposium on pure mathematics vol 65 american mathematical society providence ri 1999 about the symmetry for the ginzburg landau system in the case n m 3 we also formulate three open problems concerning the classification of entire solutions of the ginzburg landau system in any dimension

higher order quasiconvexity reduces to quasiconvexity in this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions precisely it is proved that a smooth strictly 2 quasiconvex function with p growth at infinity p 1 is the restriction to symmetric matrices of a 1 quasiconvex function with the same growth as a consequence lower semicontinuity results for second order variational problems are deduced as corollaries of well known first order theorems

a γ convergence result for the two gradient theory of phase transitions the generalization to gradient vector fields of the classical double well singularly perturbed functionals i e u 1w u e 2 u 2 dx where w 0 if and only if a or b and a b is a rank 1 matrix is considered under suitable constitutive and growth hypotheses on w it is shown that i e converge to k h n 1 s u if u w 1 1 r d i u u bv a b otherwise where k is the constant interfacial energy per unit area

brief h control and filtering of discrete time stochastic systems with multiplicative noise linear discrete time systems with stochastic uncertainties in their state space matrices are considered the problems of finite horizon filtering and output feedback control are solved taking into account possible cross correlations between the uncertain parameters in both problems a cost function is defined which is the expected value of the relevant standard h performance index with respect to the uncertain parameters a solution to the filtering problem is obtained first by applying the adjoint system and deriving a bounded real lemma for this system this solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error the solution of the filtering problem is also extended to the infinite horizon case the results of the filtering problem are used to solve the corresponding output feedback problem a filtering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques

quantization and motion law for ginzburg landau vortices we study the vortex trajectories for the two dimensional complex parabolic ginzburg landau equation without a well preparedness assumption we prove that the trajectory set is rectifiable and satisfies a weak motion law in the case of degree 1 vortices the motion law is satisfied in the classical sense moreover dissipation occurs only at a finite number of times away from these times possible collisions and splittings of vortices are constrained by algebraic equations involving their topological degrees

rigidity and gamma convergence for solid solid phase transitions with so 2 invariance the singularly perturbed two well problem in the theory of solid solid phase transitions takes the form i e u 1 ew u e 2 u 2 where u r n r n is the deformation and w vanishes for all matrices in k so n a u so n b we focus on the case n 2 and derive by means of gamma convergence a sharp interface limit for i e the proof is based on a rigidity estimate for low energy functions our rigidity argument also gives an optimal two well liouville estimate if u has a small bv norm compared to the diameter of the domain then in the l 1 sense either the distance of u from so 2 a or the one from so 2 b is controlled by the distance of vu from k this implies that the oscillation of vu in weak l1 is controlled by the l1 norm of the distance of u to k

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

a finite integration method for conformal structured grid electromagnetic simulation we describe a numerical scheme for solving maxwell s equations in the frequency domain on a conformal structured non orthogonal multi block mesh by considering maxwell s equations in a volume parameterized by dimensionless curvilinear coordinates we obtain a set of tensor equations that are a continuum analogue of common circuit equations and that separate the metrical and metric free parts of maxwell s equations and the material constitutive relations we discretize these equations using a new formulation that treats the electric field and magnetic induction using simple basis function representations to obtain a discrete form of faraday s law of induction but that uses finite integral representations for the displacement current and magnetic field to obtain a discrete form of ampere s law as in the finite integration technique t weiland a discretization method for the solution of maxwell s equations for six component fields electron commun ae u 31 1977 116 t weiland time domain electromagnetic field computation with finite difference methods int j numer model electron netw dev field 9 1996 295 319 we thereby derive new projection operators for the discrete tensor material equations and obtain a compact numerical scheme for the discrete differential operators this scheme is shown to exhibit significantly reduced numerical dispersion when compared to the standard linear finite element method we take advantage of the mesh structure on a block by block basis to implement these numerical operators efficiently and achieve computational speed with modest memory requirements when compared to explicit sparse matrix storage using the jacobi davidson g l g sleijpen h a van der vorst a jacobi davidson iteration method for linear eigenvalue problems siam j matrix anal appl 17 2 1996 401 425 s j cooke b levush eigenmode solution of 2 d and 3 d electromagnetic cavities containing absorbing materials using the jacobi davidson algorithm j comput phys 157 1 2000 350 370 and quasi minimal residual r w freund n m nachtigal qmr a quasi minimal residual method for non hermitian linear systems numer math 60 1991 315 339 iterative matrix solution algorithms we solve the resulting discrete matrix eigenvalue equations and demonstrate the convergence characteristics of the algorithm we validate the model for three dimensional electromagnetic problems both cavity eigenvalue solutions and a waveguide scattering matrix calculation

linear quadratic regulation and stabilization of discrete time systems with delay and multiplicative noise this paper is concerned with the long standing problems of linear quadratic regulation lqr control and stabilization for a class of discrete time stochastic systems involving multiplicative noises and input delay these fundamental problems have attracted resurgent interests due to development of networked control systems an explicit analytical expression is given for the optimal lqr controller more specifically the optimal lqr controller is shown to be a linear function of the conditional expectation of the state with the feedback gain based on a riccati zxl difference equation it is also shown that the system is stabilizable in the mean square sense if and only if an algebraic riccati zxl equation has a particular solution these results are based on a new technical tool which establishes a non homogeneous relationship between the state and the costate of this class of systems and the introduction of a new lyapunov function for the finite horizon optimal control design

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

global minimizers for a p ginzburg landau type energy in r 2 given a p 2 we prove existence of global minimizers for a p ginzburg landau type energy over maps on r2 with degree d 1 at infinity for the analogous problem on the half plane we prove existence of a global minimizer when p is close to 2 the key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d 1 from an arbitrary map of degree d 1 without increasing the p ginzburg landau energy

curl free ginzburg landau vortices for certain nonlinear elliptic pde problems in two dimensions the classical isoperimetric inequality produces a sharp inequality that violates a pohozaev identity except for radial symmetric decreasing solutions a generalized version of this technique is used here to prove radial symmetry of curl free ginzburg landau vortices

morrey campanato estimates for helmholtz equations abstract we derive uniform weighted l 2 and morrey campanato type estimates for helmholtz equations in a medium with a variable index which is not necessarily constant at infinity our technique is based on a multiplier method with appropriate weights which generalize those of morawetz for the wave equation we also extend our method to the wave equation

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

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

on the structure of fractional degree vortices in a spinor ginzburg landau model abstract we consider a ginzburg landau functional for a complex vector order parameter whose minimizers exhibit vortices with half integer degree by studying the associated system of equations in r 2 which describes the local structure of these vortices we show some new and unconventional properties of these vortices in particular one component of the solution vanishes but the other does not we also prove the existence and uniqueness of equivariant entire solutions and provide a second proof of uniqueness valid for a large class of systems with variational structure

minimization of a quasi linear ginzburg landau type energy abstract let g be a smooth bounded domain in r 2 consider the functional e e u 1 2 g p 0 t x k u l u 2 1 4 e 2 g 1 u 2 2 on the set h g 1 g c u h 1 g c u g on g where g is a given boundary data with degree d 0 in this paper we will study the behavior of minimizers u e of e e and we will estimate the energy e e u e

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

structure and rigidity of functions in bv2loc r2 with gradients taking only 3 values consider a function u bv2loc r2 such that u takes values in a fixed set of 3 vectors almost everywhere this condition implies that u is piecewise affine away from a closed set of vanishing 1 dimensional hausdorff measure furthermore there is some rigidity in the sense that away from the exceptional set small perturbations of u will result only in controllable changes of the structure

non trivial l q solutions to the ginzburg landau equation it is shown that the contour problem for the stationary ginzburg landau equation where x r with r x is well posed in l 4 n for a class of small data f l 2 s n 1

control for ito stochastic systems with input delay this paper examines the long standing problem of linear quadratic regulation and stabilization for ito stochastic systems with input delay this problem remains a primary challenge because the separation principle does not hold for ito stochastic systems this paper presents a complete solution to the problem 1 the sufficient and necessary solvability condition of the optimal control and the analytical controller are given based on the modified riccati differential equation defined herein 2 the sufficient and necessary stabilization condition in mean square sense is explored we show that the ito stochastic system with input delay is stabilized if and only if the modified algebraic riccati equation developed in this paper has a unique positive definite solution the essential obstacle encountered in this paper concerns a delayed forward backward stochastic differential equation d fbsde which is mathematical challenging

on lower semicontinuity of a defect energy obtained by a singular limit of the ginzburg landau type energy for gradient fields a defect energy j which measures jump discontinuities of a unit length gradient field is studied the number indicates the power of the jumps of the gradient fields that appear in the density of j it is shown that j for 3 is lower semicontinuous on the space of unit gradient fields belonging to bv in l 1 convergence of gradient fields a similar result holds for the modified energy which measures only a particular type of defect the result turns out to be very subtle since with 3 is not lower semicontinuous as is shown in this paper the key idea behind semicontinuity is a duality representation for j 3 and the duality representation is also important for obtaining a lower bound by using j 3 for the relaxation limit of the ginzburg landau type energy for gradient fields the lower bound obtained here agrees with the conjectured value of the relaxation limit

receding horizon control for multiplicative noise stochastic systems with input delay abstract this paper is concerned with finite horizon stabilization control for a class of discrete time stochastic systems subject to multiplicative noise and input delay by constructing a new cost function a complete solution to the problem of finite horizon stabilization is given for the first time based on previous work zhang et al 2015 it is shown that the system can be stabilized in the mean square sense with the receding horizon control rhc if and only if two new inequalities on the terminal weighting matrices are satisfied moreover the two inequalities can be solved by using iterative algorithm the explicit stabilizing controller is derived by solving a finite horizon optimal control problem simulations demonstrate the effectiveness of the proposed method

k quasi convexity reduces to quasi convexity the relation between quasi convexity and k quasiconvexity k greater than or equal to 2 is investigated it is shown that every smooth strictly k quasi convex integrand with p growth at infinity p 1 is the restriction to kth order symmetric tensors of a quasiconvex function with the same growth when the smoothness condition is dropped it is possible to prove an approximation result as a consequence lower semicontinuity results for kth order variational problems are deduced as corollaries of well known first order theorems this generalizes a previous work by dal maso et al in which the case where k 2 was treated

local minimizers for the ginzburg landau energy whose critical points in h 1 g u h 1 r2 u g correspond precisely to solutions of problem 0 1 the asymptotic behavior of these solutions as e 0 has attracted considerable interest in recent years bethuel brezis and helein in 1 have described in detail the limit as e 0 of the global minimizers of 0 2 in the case starshaped we describe next their result let en be any sequence approaching zero and un be a global minimizer of 0 2 for e en then un has a subsequence which has a limit u which in complex form becomes