low frequency behavior of solutions to electromagnetic scattering problems in chiral media this paper deals with the low frequency scattering problem by a conducting object in a chiral medium using boundary integral equation methods we obtain the limit of the scattered field solution of the drude born fedorov equations as frequency tends to zero making use of a hodge decomposition of the tangent fields shows that the asymptotics strongly depend on the topological properties of the domains under consideration the asymptotic expansion s dependence on the chirality parameter is shown completely

critical behaviour of a fluid in a random shear flow renormalization group analysis of a simplified model critical behaviour of a fluid binary mixture or liquid crystal subjected to strongly anisotropic turbulent mixing is studied by means of the field theoretic renormalization group as a simplified model relaxational stochastic dynamics of a non conserved scalar order parameter coupled to a random velocity field with prescribed statistics is considered the velocity is taken gaussian white in time with a correlation function of the form t t k d where k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda 1990 commun math phys 131 381 within the context of passive scalar advection it is shown that depending on the relation between the exponent and the space dimensionality d the system exhibits various types of large scale self similar behaviour associated with different infrared attractive fixed points of the renormalization group equations in addition to well known asymptotic regimes model a of equilibrium critical dynamics and a passively advected scalar with no self interaction the existence of a new non equilibrium and strongly anisotropic type of critical behaviour universality class is established and the corresponding critical dimensions are calculated to the second order of the double expansion in and e 4 d two loop approximation the most realistic values of the model parameters for example d 3 and the kolmogorov exponent 4 3 belong to this class the scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different the results are in qualitative agreement with the results obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows

the inverse spectral problem for self adjoint hankel operators in 8 5 we have considered a geometric problem in the theory of stationary gaussian processes and we have reduced this problem to the problem of the description of the bounded linear operators on hilbert space that are unitarily equivalent to moduli of hankel operators in this chapter we are going to solve the latter problem which in turn will lead to a solution of the above geometric problem in prediction theory

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

on a selfadjoint realization of curl in exterior domains the first order vectorial operator curl occurs in many areas of applications e g in electrodynamics fluid mechanics hydrodynamics magnetohydrodynamics and elasticity theory it is the purpose of this note to discuss a particular instance of curl where a domain can be given to curl so that it becomes a selfadjoint operator the particular domain of selfadjointness under consideration is singled out by its relevance in applications to establish this operator we initially consider curl as an operator curl c read restricted to in l2 defined by curl c c l2 l2 0 1

nonlinear time harmonic maxwell equations in an anisotropic bounded medium abstract we find solutions e r 3 of the problem x 1 e 2 e x e e f x e in e 0 on on a bounded lipschitz domain r 3 with exterior normal r 3 here denotes the curl operator in r 3 the equation describes the propagation of the time harmonic electric field e x e i t in an anisotropic material with a magnetic permeability tensor x r 3 3 and a permittivity tensor e x r 3 3 the boundary conditions are those for surrounded by a perfect conductor it is required that x and e x are symmetric and positive definite uniformly for x and that e l r 3 3 the nonlinearity f r 3 r is superquadratic and subcritical in e the model nonlinearity being of kerr type f x e x e p for some 2 p 6 with x g l 3 invertible for every x and 1 l r 3 3 we prove the existence of a ground state solution and of bound states if f is even in e moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries

an electromagnetic inverse problem in chiral media we consider the inverse boundary value problem for maxwell s equations that takes into account the chirality of a body in r3 more precisely we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters namely the conductivity electric permittivity magnetic permeability and chirality in the interior we rewrite maxwell s equations as a first order perturbation of the laplacian and construct exponentially growing solutions and obtain the result in the spirit of complex geometrical optics

spectral theory of piecewise continuous functions of self adjoint operators let h0 h be a pair of self adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true we give an explicit description of the absolutely continuous spectrum of the operator d h h0 for piecewise continuous functions this description involves the scattering matrix for the pair h0 h evaluated at the discontinuities of we also prove that the singular continuous spectrum of dis empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the thresholds of the absolutely continuous spectrum of d our approach relies on the construction of model operators for each jump of the function these model operators are defined as certain symmetrised hankel operators which admit explicit spectral analysis we develop the multichannel scattering theory for the set of model operators and the operator h h0 as a by product of our approach we also construct the scattering theory for general symmetrised hankel operators with piecewise continuous symbols

an elementary proof for a compact imbedding result in generalized electromagnetic theory propriete de plongement compact et certaines de ses consequences on etudie la validite du plongement suivant r 9 g e q 1 d q g l 2 q loc g

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

sign of corrections to scaling amplitudes field theoretic considerations and results for self repelling walks i discuss the application of renormalized field theory to such critical systems where the renormalized coupling approaches the fixed point from the wrong strong coupling side in contrast to a belief sometimes expressed in the literature i find that also this situation can be described within the standard formalism only the interpretation of the strong coupling branch relies on the existence of a finite cutoff i illustrate these considerations with an analysis of monte carlo data for self avoiding walks finding that the data support the theory in all respects however it is stressed that the nonuniversal parameters of the renormalized theory in general have no simple relation to the parameters of the underlying bare model

poincare meets korn via maxwell extending korn s first inequality to incompatible tensor fields for a bounded domain r3 r3 with lipschitz boundary and some relatively open lipschitz subset t t of we prove the existence of some c 0c 0 such that equation 0 1 c t l2 r3 3 symt l2 r3 3 curlt l2 r3 3 holds for all tensor fields in h curl h curl i e for all square integrable tensor fields t r3 3t r3 3 with square integrable generalized rotation curlt r3 3 having vanishing restricted tangential trace on t t if t t 0 1 still holds at least for simply connected and for all tensor fields t h curl t h curl which are l2 l2 perpendicular to so 3 so 3 i e to all skew symmetric constant tensors here both operations curl and tangential trace are to be understood row wise for compatible tensor fields t vt v 0 1 reduces to a non standard variant of the well known korn s first inequality in r3r3 namely c v l2 r3 3 sym v l2 r3 3 for all vector fields v h1 r3 v h1 r3 for which vn vn n 1 3n 1 3 are normal at t t on the other hand identifying vector fields v h1 r3 v h1 r3 having the proper boundary conditions with skew symmetric tensor fields t 0 1 turns to poincare s inequality since 2c v l2 r3 c t l2 r3 3 curlt l2 r3 3 2 v l2 r3 therefore 0 1 may be viewed as a natural common generalization of korn s first and poincare s inequality from another point of view 0 1 states that one can omit compatibility of the tensor field t at the expense of measuring the deviation from compatibility through curl t decisive tools for this unexpected estimate are the classical korn s first inequality helmholtz decompositions for mixed boundary conditions and the maxwell estimate

on spectral theory for schrodinger operators with operator valued potentials abstract given a complex separable hilbert space h we consider differential expressions of the type d 2 d x 2 v x with x a or x r here v denotes a bounded operator valued potential v b h such that v is weakly measurable and the operator norm v b h is locally integrable we consider self adjoint half line l 2 realizations h in l 2 a d x h associated with assuming a to be a regular endpoint necessitating a boundary condition of the type sin u a cos u a 0 indexed by the self adjoint operator b h in addition we study self adjoint full line l 2 realizations h of in l 2 r d x h in either case we treat in detail basic spectral theory associated with h and h including weyl titchmarsh theory green s function structure eigenfunction expansions diagonalization and a version of the spectral theorem

on approximate electromagnetic cloaking by transformation media we give a comprehensive study on regularized approximate electromagnetic cloaking in the spherical geometry via the transformation optics approach the following aspects are investigated i near invisibility cloaking of passive media as well as active radiating sources ii the existence of cloak busting inclusions without lossy medium lining iii overcoming the cloaking busts by employing a lossy layer outside the cloaked region and iv the frequency dependence of the cloaking performances we address these issues and connect the obtained asymptotic results to singular ideal cloaking numerical verifications and demonstrations are provided to show the sharpness of our analytical study

nearly cloaking the electromagnetic fields the approximate cloaking is investigated for time harmonic maxwell s equations via the approach of transformation optics the problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high conducting layer sharp estimates are established in terms of the asymptotic parameter which are independent of the material tensors of the small electromagnetic inclusion the result implies that the blow up a small region construction via the transformation optics approach yields a near cloak for the electromagnetic waves a novelty lies in the fact that the geometry of the cloaking construction of this work can be very general moreover by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region arbitrary electromagnetic contents can be nearly cloaked our mathematical technique extends the general one developed in h y

inverse problem for schrodinger equations with yang mills potentials in a slab abstract in this work we consider the inverse boundary value problem for schrodinger equations with yang mills potentials in the domain of infinite slab type we prove that the potentials can be determined uniquely up to a gauge equivalent class assuming that only partial measurements are known on the boundary hyperplanes

debye sources beltrami fields and a complex structure on maxwell fields the debye source representation for solutions to the time harmonic maxwell equations is extended to bounded domains with finitely many smooth boundary components a strong uniqueness result is proved for this representation natural complex structures are identified on the vector spaces of time harmonic maxwell fields it is shown that these complex structures are uniformized by the debye source representation that is represented by a fixed linear map on a fixed vector space independent of the frequency this complex structure relates time harmonic maxwell fields to constant k beltrami fields i e solutions of the equation e k e a family of self adjoint boundary conditions are defined for the beltrami operator this leads to a proof of the existence of zero flux constant k force free beltrami fields for any bounded region in 3 as well as a constructive method to find them the family of self adjoint boundary value problems defines a new spectral invariant for bounded domains in 3 2015 wiley periodicals inc

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

solutions of the divergence operator on john domains if rn is a bounded domain the existence of solutions u w01 p of divu f for f lp with vanishing mean value and 1 p is a basic result in the analysis of the stokes equations it is known that the result holds when is a lipschitz domain and that it is not valid for domains with external cusps in this paper we prove that the result holds for john domains our proof is constructive the solution u is given by an explicit integral operator acting on f to prove that u w01 p we make use of the calderon zygmund singular integral operator theory and the hardy littlewood maximal function for domains satisfying the separation property introduced in s buckley p koskela sobolev poincare implies john math res lett 2 5 1995 577 593 and 1 p n we also prove a converse result thus characterizing in this case the domains for which a continuous right inverse of the divergence exists in particular our result applies to simply connected planar domains because they satisfy the separation property

steady vortex patches near a rotating flow with constant vorticity in a planar bounded domain abstract in this paper we study the possible equilibrium states of the 2d incompressible euler equations in a bounded domain by solving a suitable minimization problem we prove existence of a new type of steady euler flows in which the vorticity vanishes on a finite number of compact sets of small diameter and is a positive constant elsewhere the locations of those small compact sets are determined by the geometry of the domain

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

on inequalities of korn friedrichs and babuska aziz korn s inequalities for integrals of quadratic functionals subject to certain side conditions have played a fundamental role in the development of elasticity theory see e g i 14 the korn inequalities are essential in establishing coerciveness of the differential operators of linear elastostatics and thus form the basis for existence results in that theory 9 among other areas in elasticity where consideration of korn s inequalities arise we cite for example fundamental studies on the mathematical foundations of finite elements 15 17 stability theory 11 and qualitative analyses of solution behavior such as those involved in the investigation of saint venant s principle 18 20 in the latter two areas of application in particular information on the optimal constants appearing in the inequalities the korn constants is of importance thus several investigations have been concerned with evaluating the korn constants for specific domains and with obtaining bounds for these constants 4 6 8 12 13 our purpose in this paper is to draw attention to an equivalence for two dimensional simply connected domains between korn s inequality in the second case see w 2 here and two other inequalities for quadratic functionals which have received comparatively little attention in the literature the first of these is an inequality due apparently to k o frmorichs 2 between the real and imaginary parts of an analytic function or equivalently between a harmonic function and its conjugate defined on a two dimensional domain r thus if w z f ig z xl ix2 is analytic on r friedrichs inequality reads

on the low frequency asymptotics in electromagnetic theory on considere les equations de maxwell harmoniques par rapport au temps avec des conditions aux limites on etudie le comportement asymptotique a basse frequence

multiscale asymptotic method for maxwell s equations in composite materials in this paper we discuss the multiscale analysis of maxwell s equations in composite materials with a periodic microstructure the new contributions in this paper are the determination of higher order correctors and the explicit convergence rate for the approximate solutions see theorem 2 3 consequently we present the multiscale finite element method and derive the convergence result see theorem 4 1 the numerical results demonstrate that higher order correctors are essential for solving maxwell s equations in composite materials

crossover effects in dilute magnetic materials by finite time dynamics method abstract crossover phenomena are ubiquitous in disordered system however the crossover effects from the competition between different fixed points make it hard to identify the asymptotic scaling regime controlled by the random fixed point therefore the analyses on the crossover effects are of great importance to understand phase transition in dilute magnetic materials in this paper we derive the scaling function characterizing the crossovers to probe critical behavior according to these we show an alternative method to locate the asymptotic critical region in the quenched random systems in the finite time dynamics frame for the three dimensional bond diluted ising model the asymptotic critical exponents are identified with 0 686 1 0 356 6 0 057 3 1 345 20 z 2 170 11 at middle bond concentration p 0 7 from the effective ones describing the approach to the asymptotic regime for weak dilution p 0 9 and strong dilution p 0 5 the dynamic critical exponent from our non equilibrium simulation supports distinctly z 2 18 as suggested previously as opposed to the existing larger values

inverse problems for the anisotropic maxwell equations we prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time harmonic maxwell equations in certain anisotropic settings we give a uniqueness result in the inverse problem for maxwell equations on an admissible riemannian manifold and a uniqueness result for maxwell equations in euclidean space with admissible matrix coefficients the proofs are based on a new fourier analytic construction of complex geometrical optics solutions on admissible manifolds and involve a proper notion of uniqueness for such solutions

model c critical dynamics of disordered magnets the critical dynamics of model c in the presence of disorder is considered it is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems however couplings between order parameter and secondary density cause considerable effects on non asymptotic critical properties here a general procedure for a renormalization group treatment is proposed already the one loop approximation gives a qualitatively correct picture of the diluted model c dynamical criticality a more quantitative description is achieved using two loop approximation in order to get reliable results resummation technique has to be applied

critical dynamics of the three dimensional ising model a monte carlo study extensive monte carlo simulations have been performed to analyze the dynamical behavior of the three dimensional ising model with local dynamics we have studied the equilibrium correlation functions and the power spectral densities of odd and even observables the exponential relaxation times have been calculated in the asymptotic one exponential time region we find that the critical exponentz 2 09 0 02 characterizes the algebraic divergence with lattice size for all observables the influence of scaling corrections has been analyzed we have determined integrated relaxation times as well their dynamical exponentzint agrees withz for correlations of the magnetization and its absolute value but it is different for energy correlations we have applied a scaling method to analyze the behavior of the correlation functions this method verifies excellent scaling behavior and yields a dynamical exponentzscal which perfectly agrees withz

effects of mixing and stirring on the critical behaviour stochastic dynamics of a nonconserved scalar order parameter near its critical point subject to random stirring and mixing is studied using the field theoretic renormalization group the stirring and mixing are modelled by a random external gaussian noise with the correlation function t t k4 d y and the divergence free due to incompressibility velocity field governed by the stochastic navier stokes equation with a random gaussian force with the correlation function depending on the relations between the exponents y and y and the space dimensionality d the model reveals several types of scaling regimes some of them are well known model a of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow but there are three new non equilibrium regimes universality classes associated with new nontrivial fixed points of the renormalization group equations the corresponding critical dimensions are calculated in the two loop approximation second order of the triple expansion in y y and e 4 d

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

electromagnetic scattering by a homogeneous chiral obstacle boundary integral equations and low chirality approximations time harmonic electromagnetic waves are scattered by a homogeneous chiral obstacle the corresponding transmission problem is reduced to a pair of coupled integral equations over s where s is the interface between the obstacle and the surrounding medium this is done using a generalization of the stratton chu representation that is valid for chiral media the integral equations obtained are a generalization of those obtained by muller for a homogeneous dielectric obstacle finally we develop approximations for low chirality obstacles these approximations can be computed using simple modifications to existing codes for solving muller s equations

a finite element method for approximating electromagnetic scattering from a conducting object we provide an error analysis of a fully discrete finite element fourier series method for approximating maxwell s equations the problem is to approximate the electromagnetic field scattered by a bounded inhomogeneous and anisotropic body the method is to truncate the domain of the calculation using a series solution of the field away from this domain we first prove a decomposition for the poincare steklov operator on this boundary into an isomorphism and a compact perturbation this is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem using this decomposition and edge elements to discretize the interior problem we prove an optimal error estimate for the overall problem

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

exact low frequency analysis for a class of exterior boundary value problems for the reduced wave equation in two dimensions abstract the order of convergence for the low frequency asymptotics of exterior boundary value problems for the helmholtz equation with variable possibly non smooth coefficients in two dimensions is shown to be 2 the square of the frequency except for some singular cases in these cases the asymptotics are characterized completely by some lower order terms in the spherical harmonics expansion of the solution to the static problem

on the strong unique continuation principle for inequalities of maxwell type our goal of this paper is to prove carleman type inequalities for solutions a of maxwell type inequalities curl a div a cre 1 a assuming ij ec 0 1 in a neighborhood of 0 such inequalities also arise by nonlinear maxwell equations with lipschitz nonlinearity f a provided that a is locally bounded our investigation was inspired by the study of the corresponding problem for the elliptic dirac system

critical exponents of the dilute ising model from four loop expansions four loop expansions of the coefficients of the callan symanzik equation for 3d and 2d dilute ising models are calculated fixed point coordinates and critical exponents are estimated by two methods of summation of double series a generalisation of the pade borel approximation and the first confluent form of the e algorithm of wynn summation of the double series for the 2d dilute ising model gives exponents very close to exponents of the pure ising model in accordance with the exact solution

on a structural observation in generalized electromagnetic theory are satisfied equations 0 2 describe the electromagnetic properties of the medium where 0 1 is considered if this medium is confined to a certain region g of r3 then the electromagnetic properties of its complent cc are usually represented via boundary conditions on dg we shall deal here with the boundary condition appearing if cc is assumed to be a perfect conductor i e nxe t o on c7g 13 0 0 3

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

spectral theory of discontinuous functions of self adjoint operators and scattering theory in the smooth scattering theory framework we consider a pair of self adjoint operators h0 h and discuss the spectral projections of these operators corresponding to the interval 1 the purpose of the paper is to study the spectral properties of the dierence d of these spectral projections we completely describe the absolutely continuous spectrum of the operator d in terms of the eigenvalues of the scattering matrix s for the operators h0 and h we also prove that the singular continuous spectrum of the operator d is empty and that its eigenvalues may accumulate only at thresholds in the absolutely continuous spectrum

resonance phenomena in cylindrical waveguides in the following we study aperiodic wave processes generated by periodic forces in infinite tubes consider the n dimensional cylinder q sz x co co where q is a bounded n 1 dimensional domain with smooth boundary xt sections 24 of this paper are devoted to the initial and boundary value problem af a u fe i in q x 0 a 1 1 u o on an 1 2 44 0 u x a t 0 u x for xe52 1 3 with prescribed data f u0 and u1 belonging to c q we are mainly interested in the asymptotic behavior of the solution u x t as t co our results are of the following type if w2 does not coincide with one of the eigenvalues a a of the dirichlet problem a a p v w o in q v o on a521 1 4

new integral estimates for deformations in terms of their nonlinear strains if u is a bi lipschitzian deformation of a bounded lipschitz domain in ln n 2 we show that the lp norm p 1 p n of a certain nonlinear strain function e u associated with u dominates the distance in lq q np n p if p n from u to a suitably chosen rigid motion of n this work extends that of f john who proved corresponding estimates for p 1 under the hypothesis that u has uniformly small strain we also obtain a bound for the oscillation of du in l2 these estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u in 3 the integral int limits omega e u 2d 3 is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber thus the case n 3 p 2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work

on a generalisation of krein s example abstract we generalise a classical example given by krein in 1953 we compute the difference of the resolvents and the difference of the spectral projections explicitly we further give a full description of the unitary invariants i e of the spectrum and the multiplicity moreover we observe a link between the difference of the spectral projections and hankel operators

an exterior boundary value problem for the maxwell equations with boundary data in a sobolev space we treat the time harmonic maxwell equations in an exterior domain with prescribed boundary data n e in the sobolev space of square integrable tangential fields with square integrable surface divergence by using boundary integral equation methods existence and uniqueness results are established furthermore we investigate the completeness of electric and magnetic dipoles distributed on an inner surface in this sobolev space

on an inverse boundary problem arising in brain imaging abstract we consider the inverse problem of recovering both an unknown electric current and the surrounding electromagnetic parameters of a medium from boundary measurements this inverse problem arises in brain imaging we show that under generic conditions one can recover both the source and the electromagnetic parameters if these are piecewise constant and the source current is invariant in a fixed direction or a harmonic function respectively

critical dynamics a field theoretical approach we review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of halperin and hohenberg we unify the presentation of the theoretical background by restricting ourselves to the field theoretic renormalization group method the main results obtained in the different universality classes are presented this contains the critical dynamics near the gas liquid transition in pure fluids model h the plait point and consolute point in mixtures model h the superfluid transition in 4 he model f and 4 he 3 he mixtures model f the curie point model j and neel point model g in heisenberg magnets and the superconducting transition in comparison with experimental results it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non universal effective behaviour either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics e g at the superfluid transition or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas liquid or demixing transition the calculation of the critical dynamics is adequate in most cases only in two loop order we review these results and present the solution to unreasonable features found for some models thus we consider model c where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model model a have not been understood in general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role their fixed point values and the approach to the fixed point are changed considerably in two loop order compared to their values in one loop order these considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium model f and the isotropic antiferromagnet model g as far as possible the comparison of results obtained by the renormalization group theory with numerical simulations has been made

electromagnetic inverse problems and generalized sommerfeld potentials in this article we study the inverse boundary value problem for maxwell s equation that is we determine the scalar permeability permissibility and conductivity of a body from electromagnetic field measurements outside the body in this paper the problem is approached by using certain generalized sommerfeld potentials that are shown to satisfy a vector valued schrodinger equation by using these potentials the inverse boundary value problem is reduced to the problem of determining the potential of this second order operator further it is shown that the electromagnetic material parameters can be determined by using electric or magnetic dipole point sources only

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

existence of minimizers for a finite strain micromorphic elastic solid we investigate geometrically exact generalized continua of micromorphic type in the sense of eringen the two field problem for the macrodeformation and the affine microdeformation p gl 3 r in the quasistatic conservative load case is investigated in a variational form depending on material constants two existence theorems in sobolev spaces are given for the resulting nonlinear boundary value problems these results comprise existence results for the micro incompressible case p sl 3 r and the cosserat micropolar case p so 3 r in order to treat external loads a new condition called bounded external work has to be included which overcomes the conditional coercivity of the formulation the possible lack of coercivity is related to fracture of the micromorphic solid the mathematical analysis uses an extended korn first inequality the methods of choice are the direct methods of the calculus of variations

the dynamic critical exponent of dilute and pure ising systems a massive dynamic field theory is employed to calculate the critical exponent z of the three dimensional dilute ising model to two loop order without expansion in square root 4 d with result z approximately 2 191 applying the same method to the pure ising model at three loop order we find z 2 022 for d 3 and z 2 124 for d 2

boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle time harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle the corresponding electromagnetic transmission problem is reduced to a single integral equation over s for a single unknown tangential vector field where s is the interface between the obstacle and the surrounding medium in fact several different integral equations are derived and analysed including two previously known equations due to e marx and j r mautz and two new singular integral equations mautz s equation is shown to be uniquely solvable at all frequencies a new uniquely solvable singular integral equation is also found the paper also includes a review of methods using pairs of coupled integral equations over s it is these methods that are usually used in practice although single integral equations seem to offer some computational advantages

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

critical exponents of random ising like systems in general dimensions critical exponents of weakly dilute ising like systems are computed for non integer space dimensionalities in the range 2 d 4 the calculations are performed in the framework of the callan symanzik field theoretic approach two loop renormalization group functions are obtained as renormalized perturbation theory series expansions directly in noninteger dimensions the values of the critical exponents are estimated with the use of the two variable borel resummation method

domain dependent dirac s delta and derivatives with an application to electromagnetic boundary integral representations the domain dependent versions of derivatives and dirac s delta are defined in distributional sense these operations enable to obtain domain dependent fundamental solutions and global boundary integral representation formulae a global representation formula is defined everywhere also on the boundary and includes the jump relations of the boundary the use of the domain dependent objects can be interpreted as taking the boundary limit in prior to integrating by parts when deriving the familiar boundary integral equations as an application the representation formulae are obtained for the solutions of the helmholtz equation and the maxwell equations

aperiodic electromagnetic waves in exterior domains on considere les problemes aux valeurs limites et initiales pour l equation d onde vectorielle dans des domaines exterieurs par rapport aux conditions limites electriques n e 0 e 0 sur et des conditions aux limites magnetiques n h 0 n h 0 sur on etudie le comportement asymptotique des solutions e x t et h x t quand

scattering with critically singular and delta shell potentials we consider a scattering problem for electric potentials that have a component which is critically singular in the sense of lebesgue spaces and a component given by a measure supported on a compact lipschitz hypersurface we study direct and inverse point source scattering under the assumptions that the potentials are real valued and compactly supported to solve the direct scattering problem we introduce two functional spaces sort of bourgain type spaces that allow to refine the classical resolvent estimates of agmon and hormander and kenig ruiz and sogge these spaces seem to be very useful to deal with the critically singular and delta shell components of the potentials at the same time furthermore these spaces and their corresponding resolvent estimates turn out to have a strong connection with the estimates for the conjugated laplacian used in the context of the inverse calderon problem in fact we derive the classical estimates by sylvester and uhlmann and the more recent ones by haberman and tataru after some embedding properties of these new spaces regarding the inverse scattering problem we prove uniqueness for the potentials from point source scattering data at fix energy to address the question of uniqueness we combine some of the most advanced techniques in the construction of complex geometrical optics solutions

on the critical fluctuations in superconductors the field theoretical gauge model for a superconductor generalized to an n 2 component complex order parameter is considered the question of the order of the phase transition occurring in this model is discussed previous renormalization group calculations suggest that in such a model a fluctuation induced first order phase transition occurs we re examine previously obtained expressions for the renormalization group functions in the two loop approximation in three dimensions special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence we discuss the possible ways of analytical continuation of the series obtained comparing results obtained by direct calculations with those obtained by pade analysis and the pade borel resummation technique the conjecture is made that in the model under consideration there still exists a possibility for a second order phase transition

absolutely continuous spectrum of dirac operators for long range potentials abstract using the resolvent method and the technique of weighted l2 estimates we deduce the non existence of the singular continuous spectrum of the dirac operator p x on the interval 1 we assume that the hermitian matrix potential p x p1 x p2 x is divided into a long range part p 1 x o x e r r p 1 x o x e and a short range part p 2 x o x 1 e x with local singularities p x o x a j 1 of coulomb type for n nuclei a1 an r 3

on korn s first inequality with non constant coefficients in this paper we prove a korn type inequality with non constant coefficients which arises from applications in elasto plasticity at large deformations more precisely let r 3 be a bounded lipschitz domain and let be a smooth part of the boundary with non vanishing two dimensional lebesgue measure define and let be given with det f p x 0 moreover suppose that rot then clearly this result generalizes the classical korn s first inequality which is just our result with f p 11 with slight modifications we are also able to treat forms of the type

an initial boundary value problem for the maxwell equations abstract in this paper by using methods from complex analysis and quaternionic analysis we investigate an initial boundary value problem for the maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases in addition by using a similar method we also discuss an initial boundary value problem for a hyperbolic complex system of first order equations in r 3

weak solution theory for maxwell s equations in the semistatic limit case we consider the quasilinear semistatic system of the reduced maxwell s equations curl h j curl e b 0 div b 0 for the electromagnetic field given by the electric field e and the magnetic induction b in a bounded domain these equations result from the usual system of maxwell s equations by neglecting displacement currents the domain is permitted to have arbitrary topological genus and a nonsmooth lipschitz type boundary the constituent relations for the underlying medium occupying are j e linear h b nonlinear here is a strongly monotone and lipschitz continuous map in the hilbert space of vector valued functions with components in l2 the electromagnetic field is subject to generalized boundary conditions associated with the classical form n e n b 0 on we show global existence and uniqueness and continuous dependence on the data of the weak solution of the associated initial boundary value problem in a suitable space time hilbert space setting uniqueness is proven directly the existence result is derived by reduction to an abstract evolution equation of monotone type in an appropriately constructed gelfand triplet via the introduction of electromagnetic potentials we solve the evolution by means of a classical galerkin approximation method and then show that the potentials so obtained do define the desired solution of the original equations finally we prove the continuous dependence of the solutions to the original problem from their initial data

on the corrections to scaling in three dimensional ising models the leading correction to scaling amplitudes for the spin 1 2 nearest neighbor sc bcc and fee ising models are considered with the particular aim of determining their signs on the basis of previous two variable series analyses by chen fisher and nickel and renormalization group 4 d expansions it is concluded that the correction amplitudes for the susceptibility correlation length specific heat and spontaneous magnetization arenegative for all three lattices thus for example the effective exponent eff t asymptotically approaches the true susceptibility exponent fromabove other earlier and more recent high temperature series and field theoretic analyses are seen to be consistent with this result however the usual nonasymptotic perturbative field theoretic approaches are essentially committed to positive correction amplitudes the question of the signs therefore relates directly to the applicability of these non asymptotic field theoretic calculations to three dimensional ising models as well as to different experimental systems

the essential spectrum of the nonself adjoint maxwell operator in a bounded domain 1 introductionin this work we study the spectrum of the maxwell operator in theself adjoint case it is well known that the spectrum consists of oneeigenvalue with infinite multiplicity in zero and other eigenvalues havingfinite multiplicities however the spectrum of the nonself adjoint operatorwas not studied much though the nonself adjoint perturbation rises from avery common phenomena namely from the conductivity in this work weshow that the maxwell system with conductivity in a bounded domain hasquite a large essential spectrum the essential spectrum contains aninterval or another nondiscrete set let v be a

a linearized inverse boundary value problem for maxwell s equations abstract in this paper we consider the problem of determining the electromagnetic state of a body from measurements made on the surface of the body we study the full set of maxwell s equations that govern time harmonic electric and magnetic fields we seek to reconstruct the magnetic permeability the electric permitivity and the electric conductivity in the interior of the body from measurements made on the surface we exhibit appropriate boundary measurements in the form of a boundary mapping specifically the mapping from the tangential components of the electric field to the tangential components of the magnetic field this data can be used to reconstruct and approximately provided they deviate only slightly from known constants we also estimate the reconstruction errors

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

a variational principle for three dimensional water waves over beltrami flows abstract we consider steady three dimensional gravity capillary water waves with vorticity propagating on water of finite depth we prove a variational principle for doubly periodic waves with relative velocities given by beltrami vector fields under general assumptions on the wave profile

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space