a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

the radial volume derivative and the critical boundary displacement for cavitation we study the displacement boundary value problem of minimizing the total energy e mathbf u stored in a nonlinearly elastic body occupying a spherical domain b in its reference configuration over possibly discontinuous radial deformations mathbf u of the body subject to affine boundary data mathbf u mathbf x lambda mathbf x for mathbf x in partial b for a given value of lambda we define what we call the radial volume derivative at lambda denoted by g lambda which measures the stability or instability of the underlying homogeneous deformation mathbf u h lambda mathbf x equiv lambda mathbf x to the formation of holes we give conditions under which the critical boundary displacement lambda mathrm crit for radial cavitation is the unique solution of g lambda 0 moreover we prove that the radial volume derivative g lambda can be approximated by the corresponding volume derivative for a punctured ball b epsilon containing a pre existing cav

mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

generators and defining relations for the ring of differential operators on a smooth affine algebraic variety in prime characteristic abstract for the ring of differential operators d o x on a smooth affine algebraic variety x over a perfect field of characteristic p 0 explicit algebra generators and defining relations are found a finite set of generators and a finite set of defining relations are given explicitly for the module der k o x of derivations on the algebra o x of regular functions on the variety x

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

global f regularity of schubert varieties with applications to d modules a projective algebraic variety x over an algebraically closed field k of positive characteristic is called globally f regular if the section ring s c 0n o h x cn of an ample line bundle c on x is strongly f regular in the sense of h chster and huneke 9 cf definition 1 1 this important notion was introduced by karen smith in 18 in this paper we prove that schubert varieties are globally f regular an im mediate consequence is that local rings of schubert varieties are strongly f regular and thereby f rational another consequence is that local rings of varieties like de terminantal varieties that can be identified with open subsets of schubert varieties cf 13 are strongly f regular let x denote a flag variety and y c x a schubert variety over k then the local cohomology sheaves hjy ox are equivariant for the action of the borel subgroup and holonomic in the sense of 4 x x niodules as an application of ir rationality of schubert varieties we apply recent results of blickle cf 2 to prove that the simple objects in the category of equivariant and holonomic px modules are precisely the local cohomology sheaves hy ox where c is the codimension of y in x using a local grothendieck cousin complex from 11 we prove that the decomposition of the local cohomology modules with support in bruhat cells is multiplicity free see 4 2 in characteristic zero the local cohomology modules with support in bruhat cells correspond to dual verma modules in this setting the decomposition behavior and the simple t x modules arise from intersection cohomology complexes of schubert varieties by the riemann hilbert correspondence if we pick the singular codi mension one schubert variety y in the full flag variety z for sl4 in characteristic zero computations in kazhdan lusztig theory show that hy oz is not a simple z module see 4 1

lie algebra of anomalously scaled fluctuations for quantum lattice systems it is proven that anomalously scaled fluctuations have a natural lie algebra structure the harmonic lattice in the ground state is given as an illustration of the general theorem

on critical points of the σ2 energy over a space of measure preserving maps abstract let x r n be a bounded lipschitz domain and consider the 2 energy functional f 2 u x x u u 2 d x over the space of admissible maps a x u w 1 4 x r n u x x det u 1 for l n a e in x a good measure of how much a map u stretches areas of 2 dimensional sub manifolds of the domain x is the norm of u u 2 t x 2 t x analogously to u 2 the dirichlet energy density that is a measure of length s stretching these kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by t skyrme in 1961 in this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the euler lagrange equations associated with f 2 x over a x in particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper shahrokhi dehkordi and taheri 2009 17 indeed we show that in even dimensions the latter system of equations admits infinitely many smooth solutions modulo isometries amongst such maps in odd dimensions this number reduces to one the result relies on a careful analysis of the full versus the restricted euler lagrange equations we investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

non commutative central limits non commutative central limit theorems are derived the ccr c algebra of fluctuations is analyzed in detail the stability of the central limit is studied by means of the notion of relative entropy

radially symmetric motions of nonlinearly viscoelastic bodies under live loads this paper treats radially symmetric motions of nonlinearly viscoelastic circular cylindrical and spherical shells subjected to the live loads of centrifugal force and time dependent hydrostatic pressures the governing equations are exact versions of those for 3 dimensional continuum mechanics so shell does not connote an approximate via some shell theory these motions are governed by quasilinear third order parabolic hyperbolic equations having but one independent spatial variable the principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations the presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis the interaction of geometrically exact formulations the compatible use of general constitutive equations for material response and the presence of live loads show how these factors play crucial roles in the behavior of solutions in particular for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior the analysis using classical methods covers infinite time blowup for cylindrical shells subject to centrifugal forces infinite time blowup for cylindrical shells subject to steady and time dependent hydrostatic pressures finite time blowup for spherical shells subject to steady and time dependent hydrostatic pressures and the preclusion of total compression this paper concludes with a sketch using some modern methods of the existence of regular solutions until the time of blowup

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

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

spherically symmetric two phase deformations and phase transition zones this paper is concerned with characterization and stability assessment of two phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material we study general properties of equilibrium two phase spherically symmetric deformations then we specialize to phase transformations of a solid sphere that is subjected to an all round tension pressure two material models are used to demonstrate a variety of transformation behaviours and some common features for both materials we construct phase transition zones ptzs formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface then we demonstrate how the ptz can be used for the prediction of the number of two phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the ptz we show that even in the simplest case of one interface the solution is not unique two equilibrium two phase solutions as well as one uniform one phase solution are found under the same boundary conditions for the three solutions we construct their load deformation diagrams and compare the associated total energies the stability of the two phase states with respect to radial and small wavelength perturbations is also examined we observe how unstable solutions are related with the ptz

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

a numerical method for detecting singular minimizers a numerical method for computing minimizers in one dimensional problems of the calculus of variations is described such minimizers may have unbounded derivatives even when the integrand is smooth and regular in such cases because of the lavrentiev phenomenon standard finite element methods may fail to converge to a minimizer the scheme proposed is shown to converge to an absolute minimizer and is tested on an example the effect of quadrature is analyzed the implications for higher dimensional problems and in particular for fracture in nonlinear elasticity are discussed

about the exactness of the linear response theory for quantum lattice systems with finite range potentials and integrable space clustering we prove the linearity of the response theory when dealing with fluctuation observables

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

void nucleation and growth for a class of incompressible nonlinearly elastic materials abstract in this paper a bifurcation problem for a solid sphere subjected to uniform tensile dead loading p 0 at its boundary is examined within the framework of finite elastostatics the sphere is composed of a particular class of homogeneous isotropic incompressible nonlinearly elastic materials namely those of power law type one solution to the problem for all values of p 0 corresponds to a homogeneous state in which the sphere remains undeformed while stressed however for sufficiently large values of p 0 there is in addition a second possible configuration involving an internal traction free spherical cavity the dependence on constitutive parameters of the critical load at which bifurcation occurs is examined as well as the subsequent void growth the stress distribution after cavitation occurs is also described the results are obtained in closed analytic form

central limit theorems for the large spin asymptotics of quantum spins we use a generalized form of dyson s spin wave formalism to prove several central limit theorems for the large spin asymptotics of quantum spins in a coherent state

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

neohookean deformations of annuli in the higher dimensional euclidean space abstract let n 2 be an integer and assume that a x r n 1 x r and a y r n 1 y r be two annuli in euclidean space r n assume that f a a resp r a a be the class of all orientation preserving resp radial homeomorphisms h a a in the sobolev space w 1 n a a which keep the boundary circles in the same order in this paper we extended the corresponding results of iwaniec and onninen which was published in math ann vol 348 2010

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

extremal mappings of finite distortion the theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non linear elasticity and the calculus of variations in this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm for instance we show the following suppose that f o is a homeomorphism of the circle with f o 1 in cal w 1 2 2 then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non vanishing jacobian determinant the condition on f o is sharp classically the mapping f o is assumed to be quasisymmetric then there is an extremal quasiconformal mapping with boundary values f o but it is not always unique and it is seldom smooth indeed even when f o is quasisymmetric the cal l 1 minimiser for the distortion function will almost never be quasiconformal we further find that there are many new and unexpected phenomena concerning existence uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex these seem to differ markedly from phenomena observed when studying multi well type functionals

central limit theorem for locally interacting fermi gas we consider a locally interacting fermi gas in its natural non equilibrium steady state and prove the quantum central limit theorem qclt for a large class of observables a special case of our results concerns finitely many free fermi gas reservoirs coupled by local interactions the qclt for flux observables together with the green kubo formulas and the onsager reciprocity relations previously established jop4 complete the proof of the fluctuation dissipation theorem and the development of linear response theory for this class of models

hyperelastic deformations and total combined energy of mappings between annuli abstract we consider the so called combined energy of a deformation between two concentric annuli and minimize it provided that it keep order of the boundaries it is an extension of the corresponding result of euclidean energy it is interesting that the minimizers are certain radial mappings and they exist if and only if the annulus on the image domain is not too thin provided that the original annulus is fixed this in turn implies a nitsche type phenomenon next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy which also involve some nitsche type phenomenon the main part of the paper is concerned with the total combined energy a certain integral operator defined as a convex linear combination of the combined energy and combined distortion of diffeomorphisms between two concentric annuli a 1 r and b 1 r first we construct radial minimizers of total combined energy then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint this extends the main result obtained by iwaniec and onninen in 9

the nitsche conjecture the nitsche conjecture is deeply rooted in the theory of doubly connected minimal surfaces however it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli h a a r r onto a r r a in the early 1960s while attempting to describe all doubly connected minimal graphs over a given annulus a j c c nitsche observed that their conformal modulus cannot be too large then he conjectured in terms of isothermal coordinates even more a harmonic homeomorphism h a onto a exists if an only if r r 1 2 r r r r this fascinating and engaging problem remained open for almost a half of a century in the present paper we provide among further generalizations an affirmative answer to his conjecture

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

on conservation laws and necessary conditions in the calculus of variations it is well known from the work of noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law in this paper we prove that each such conservation law arises directly as the euler lagrange equation for the functional on taking suitable variations around a minimizer

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

dynamics of fluctuations for quantum lattice systems for short range interactions and forl1 space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations this dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations

irreversibility and randomness in linear response theory recall that the fluctuation dissipation theorem connects the response function of a passive linear system and the spectral density of the stationary stochastic process which describes the thermal fluctuations in the system it is shown that the classical limit 0 of the fluctuation dissipation theorem implies a correspondence between systems which are reversible in the sense that the energy used to drive them away from equilibrium is completely recoverable as work and processes which are deterministic in the sense of wiener s prediction theory while irreversible systems correspond to nondeterministic processes this correspondence is expressed by a simple transformation between the operator kernel which determines the optimal choice of the time dependent force and the linear predictor for the stochastic process for quantum systems this correspondence does not hold the fluctuations are always of the deterministic type for any finite temperature but the system is not necessarily reversible for irreversible systems a formula is derived for the instantaneous entropy production which is a generalization of the standard one for markovian dynamics

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

error analysis of a dual parametric bi quadratic fem in cavitation computation in elasticity the orientation preservation conditions and approximation errors of a dual parametric bi quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed the analytical results allow us to establish based on an error equidistribution principle an optimal meshing strategy for the method in cavitation computation numerical results are in good agreement with the analytical results

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

myriad radial cavitating equilibria in nonlinear elasticity it is shown that every bounded strictly increasing smooth positive function of sufficiently slow growth is the jacobian of a radial hole creating equilibrium deformation for an appropriately constructed compressible nonlinearly elastic energy

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

regularity of the inverse of a planar sobolev homeomorphism let be a domain suppose that f w1 1loc r2 is a homeomorphism such that df x vanishes almost everywhere in the zero set of j f we show that f 1 w1 1loc f r2 and that df 1 y vanishes almost everywhere in the zero set of sharp conditions to quarantee that f 1 w1 q f r2 for some 1 q 2 are also given

the green kubo formula and the onsager reciprocity relations in quantum statistical mechanics we study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the green kubo formula and the onsager reciprocity relations for heat fluxes generated by temperature differentials our derivation is axiomatic and the key assumptions concern ergodic properties of non equilibrium steady states

maximal smoothness of solutions to certain euler lagrange equations from nonlinear elasticity we investigate the maximal smoothness of stationary states for the multiple integral such variational problems are motivated by the study of nonlinear elasticity assuming certain structure conditions for and given a stationary state we derive an a priori l p estimate for for any p and where as a consequence we show that a c 1 stationary state necessarily satisfies det and is of class c 2 in nevertheless singular stationary states do exist we construct a nonsmooth c 1 solution for a particular in two dimensions such that det in and det vanishes at precisely one point in

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

on the symmetry of energy minimising deformations in nonlinear elasticity ii compressible materials let a bf x in mathbb r n a 0 be the region occupied by an incompressible nonlinearly hyperelastic material in its reference configuration deformations bf u it a rightarrow mathbb r n are therefore isochoric maps and so satisfy the incompressibility constraint rm det nabla bf u 1 the displacement and the mixed displacement zero traction boundary value problems are considered here a displacement boundary condition of the form bf u bf x sigma bf x is prescribed on one of the boundary components where 0 is a given constant and the displacement on the remaining boundary component is either prescribed in the same form in the case of the pure displacement boundary value problem or left unspecified in the case of the mixed boundary value problem in this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of possibly non symmetric isochoric deformations of the region a

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians