some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

an r matrix interpretation of modified systems abstract the r matrix formalism is applied to the loop algebra based on the central extension of l i v we recover by means of this single construction the modified systems of antonowicz and fordy associated with factorisations of the linear operator of the spectral problem ni 0 i ei d2 dx2 ui 0 together with a modification of ito s equation which was excluded by antonowicz and fordy we also reproduce several results of kupershmidt

twists and shear maps in nonlinear elasticity explicit solutions and vanishing jacobians in this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory we examine in particular the relationship between the positivity of the jacobian det u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps we exhibit emph explicit twist maps defined on two dimensional annuli that are stationary points of an appropriate energy functional and whose jacobian vanishes on a set of positive measure in the annulus within the class of shear maps we precisely characterize the unique global energy minimizer u r 2 in a model two dimensional case the shear map minimizer has the properties that i det u is strictly positive on one part of the domain ii det u 0 necessarily holds on the rest of and iii properties i and ii combine to ensure that u is not continuous on the whole domain

a necessary and sufficient condition for lower semicontinuity abstract it is well known that w 1 p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral i u f u x d x on the sobolev space w 1 p and that it is sufficient too provided that the integrand f satisfies suitable growth conditions related to the exponent p we show that for extended real valued integrands a closely related convexity condition defined in terms of gradient young measures is both necessary and sufficient for lower semicontinuity in the more general and flexible setting of compensated compactness our main results identify the relaxation lower semicontinuous envelope in two related situations

on the theory of relaxation in nonlinear elasticity with constraints on the determinant we consider vectorial variational problems in nonlinear elasticity of the form i u int w du rm d x where w is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero we show that under suitable growth assumptions the functional int w rm qc du rm d x is an upper bound on the relaxation of i and coincides with the relaxation if the quasiconvex envelope wqc of w is polyconvex and has p growth from below with p geqq n this includes several physically relevant examples we also show how a constraint of incompressibility can be incorporated in our results

a condition for the holder regularity of local minimizers of a nonlinear elastic energy in two dimensions we prove the local holder continuity of strong local minimizers of the stored energy functional e u int omega lambda nabla u 2 h rm det nabla u rm d x subject to a condition of positive twist the latter turns out to be equivalent to requiring that u maps circles to suitably star shaped sets the convex function h s grows logarithmically as s to 0 linearly as s to infty and satisfies h s infty if s leqq 0 these properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers the main innovation is to prove that if a local minimizer has positive twist a e on a ball then an euler lagrange type inequality holds and a caccioppoli inequality can be derived from it the claimed holder continuity then follows by adapting some well known elliptic regularity theory we also demonstrate the regularizing effect that the term int omega h rm det nabla u rm d x can have by analysing the regularity of local minimizers in the class of shear maps in this setting a more easily verifiable condition than that of positive twist is imposed with the result that local minimizers are holder continuous

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

weak lower semicontinuity of polyconvex integrals multiple integrals with polyconvex integrands are studied on the class of all sense preserving diffeomorphisms from w 1 p r n where is an open subset of r n they are proved to be sequentially weakly lower semicontinuous if 1 p n 1 an example is presented showing that a similar result is not valid if p n 1

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

regularity of relaxed minimizers of quasiconvex variational integrals with p q growth we consider autonomous integrals f u int omega f du rm d x quad rm for u mathbb r n supset omega to mathbb r n in the multidimensional calculus of variations where the integrand f is a strictly quasiconvex c2 function satisfying the p q growth conditions gamma a p leqq f a leqq gamma 1 a q quad rm for every a in mathbb r nn with exponents 1 p leqq q infty

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

minimizers for a double well problem with affine boundary conditions this paper is concerned with the existence of minimizers for functionals having a double well integrand with affine boundary conditions such functionals are related to the so called kohn strang functional which arises in optimal shape design problems in electrostatics or elasticity they are known to be not quasi convex and therefore existence of minimizers is in general guaranteed only for their quasi convex envelopes we generalize previous results in 1 and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory we also prove by a general argument that their rank one convexifications coincide with their quasi convexifications

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

on one homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions we extend a result from phillips by showing that one homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine the result is novel in two ways firstly the system is allowed to depend in a sufficiently smooth way on the spatial variable x secondly phillips s original result is shown to apply to w 1 w 2 one homogeneous solutions from which his treatment of lipschitz solutions follows as a special case a singular one homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

an application of the arnoldi s method to a geophysical fluid dynamics problem abstract a new method to find solutions of large linear systems based on a projection on the krylov subspace is shown to be successful when applied to the linearized barotropic and baroclinc primitive equations these sets of equations are widely used in the simulation of the dynamics of the atmosphere the scheme consists of projecting the original linear system on the krylov subspace thereby reducing the dimensionality of the matrix to be inverted in order to obtain the solution the iterative arnoldi s method reaches a solution even using a krylov subspace ten times smaller than the original space of the problem this generality allows us to treat the important problem of propagation of linear waves in the atmosphere from a more general point of view a larger class nonzonally symmetric of basic states can now be treated for the baroclinic primitive equations these kinds of problems leading to large unsymmetrical linear system of order 10000 or more can now be successfully tackled by the krylov approach numerical results of a general circulation model linearized around a nonsymmetrical basic state are here shown for various numbers of degrees of freedom

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

lie algebras and korteweg de vries equations abstract for each metrizable lie algebra g an infinite commuting hierarchy of bi hamiltonian integrable extensions of the korteweg de vries kdv hierarchy is found the classical kdv hierarchy itself corresponds to the case g 0 the construction is based on the hamiltonian formalism on the dual space to a certain one parameter family of differential lie algebras produced from g

partial regularity of strong local minimizers of quasiconvex integrals with p q growth defined on vector valued maps u r n 1 here denotes a bounded open set in r n 2 and f r r is a c function satisfying suitable assumptions described below the existence and the partial regularity of minimizers of f are classical issues in the modern calculus of variations and they have been extensively studied especially over the last twenty years specifically in the present paper we will focus on gradient regularity under the basic assumption that f is quasiconvex that is

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems i the algebraic structure of the gradient holonomic algorithm for lax integrable dynamical systems is discussed a generalization of the r structure approach for the case of operator valued affine lie algebras is used to prove the bi hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions the monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical lie poisson bracket on initial affine lie algebra with gauge central extension as an illustration the two dimensional operator benney kaup integrable hierarchy is considered and their bi hamiltonicity is proved

computation of the lower semicontinuous envelope of integral functionals and non homogeneous microstructures abstract a numerical method is established for computing the weakly lower semicontinuous envelope of integral functionals with non quasiconvex integrands the convergence of the method is proved and it is shown that the method is capable of capturing curved and non homogeneous microstructures numerical examples are given to show the effectiveness of the method for capturing curved and non homogeneous laminated microstructures

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

lower semicontinuity of certain quasiconvex functionals in orlicz sobolev spaces abstract let u r n r n be any vector valued function we prove a semicontinuity result in the weak topology of orlicz sobolev spaces of the functional f u f d u x d x where f is a quasiconvex function satisfying non standard growth conditions

lower semi continuity for non coercive polyconvex integrals in the limit case lower semi continuity results for polyconvex functionals of the calculus of variations along sequences of maps u n m in w 1 m 2 m n weakly converging in w 1 m 1 are established in addition for m n 1 we also consider the autonomous case for weakly converging maps in w 1 n 1

on the relaxation in bv ω rm of quasi convex integrals abstract given a quasi convex function f with linear growth we find the integral representation in bv r m of the functional f arising from the relaxation of f u f u dx u c 1 r m in the l loc 1 r m topology

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

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

variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results we use a variational convergence method to study the consistency of various cosserat hypotheses in shell theory with the limit nonlinear membrane model derived from three dimensional elasticity in the course of the analysis we introduce a generalization of quasiconvexity that is suitable for problems of the calculus of variations with two vectorial unknowns one of which appears through its gradient the other one through its value in a weak w 1 p l p framework

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

on the implementation of the gmres m method to elliptic equations in meteorology in this paper we consider the relatively new preconditioned generalized minimal residual method restarted every m iterations gmres m for the solution of three dimensional elliptic equations large sparse non symmetric matrices are involved the particular equation of interest is the quasi geostrophic omega equation often used in meteorology to compute vertical motion the gmres m method is tested with different preconditioners for the solution of two and three dimensional elliptic equations the method requires no relaxation parameters and has no restrictions on the size of the 3d grid gmres can be used for more types of matrices than other methods such as sor numerical results show that jacobi preconditioned gmres m performs best for 3d and high resolution problems among five different preconditioners tested while ilij factorization of the partial or whole matrix a as a preconditioner is good for 2d and low resolution problems the sor preconditioners for the gmres m method with optimal relaxation parameters are not as efficient and the best choice for the relaxation parameter in sor preconditioning is not the same as the best choice for the simple sor method an algorithm for using the preconditioned gmres m method is presented

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

mudpack multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations the solution regions are rectangles in two dimensions and boxes in three dimensions any combination of periodic dirichlet and mixed derivative boundary conditions is allowed the equations are automatically discretized using second order finite differencing the package will vectorize on cray computers examples are given which demonstrate ease of use efficiency and applicability to a wide range of problems

a new computational framework for electro activation in cardiac mechanics abstract this paper presents a novel computational framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle the paper presents the following main novelties 1 two new mixed formulations tailor made for active stress and active strain coupling approaches have been developed and used in conjunction with two different ionic models namely bueno orovio et al 2008 and ten tusscher et al 2004 taking as a reference the mixed formulations introduced by bonet et al 2015 in the context of nonlinear elasticity the proposed formulations include as unknown fields the geometry and the transmembrane potential and possibly a lagrange multiplier enforcing weakly the incompressibility constraint as well as the deformation gradient tensor its cofactor its determinant the gradient of the transmembrane potential and their respective work conjugates the finite element implementation of these formulations is shown in this paper where a static condensation procedure is presented in order to yield an extremely competitive computational approach 2 a comprehensive and rigorous study of different ionic models i e bueno orovio and ten tusscher and electromechanical activation couplings i e active strain and active stress has been carried out 3 an analytical and numerical analysis of the possible loss of ellipticity and polyconvexity of one of the most widely used constitutive models in the context of cardiac mechanics is carried out in this paper putting forward possible polyconvexifications of the existing model 4 in addition an invariant representation of guccione s constitutive model is proposed finally a series of numerical examples are included in order to demonstrate the applicability and robustness of the proposed formulations

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

lower semicontinuity of surface energies using the theory of indicator measures a lower semicontinuity result for quasiconvex functions in w 1 1 and assuming only l 1 convergence is obtained

some properties of the degree for a class of sobolev maps we prove that a map u w loc1 1 m n whose differential and adjoint are integrable with exponents n 1 p n and n n 1 q respectively possesses an integer valued degree with the usual properties here m c 0 1 and n c 1 1 are n dimensional oriented riemannian manifolds with m possibly non empty

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

relaxation of quasiconvex functional in bv ω ℝp for integrands f x u u in this paper it is shown that if p x u is a quasiconvex function with linear growth then the relaxed functional in bv p of u to int limits omega f x u x nabla u x dx with respect to the l1 topology has an integral representation of the form begin gathered mathfrak f u int limits omega f x u x nabla u x dx int limits sigma u k x u x u x v x dh n 1 x hfill int limits omega f infty x u x dc u hfill end gathered where du u dx u u v dhn 1l u c u the proof relies on a blow up argument introduced by fonseca muller in the case where u w1 1 and on a recent result by alberti showing that the cantor part c u is rank one valued

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer