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

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 change of subscripts of the stimulus placements for a four point sampling method because the variance parameter in a cumulative normal function is not the same as the spread parameter s of a logistic function the subscripts of the four stimulus placements of eq 3 and thereafter of lam et al j acoust soc am 99 3689 3693 1996 should be x12 x27 x73 and x88

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

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

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

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

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

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

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

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

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

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

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

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

the relaxation of non quasiconvex variational integrals we show that the steepest descent algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non quasiconvex variational integrals we introduce a multi level infinite dimensional variant of the steepest descent algorithm designed to compute complex microstructures by forming non smooth minimizers from the smooth initial guesses we apply this multi level method to the minimization of the variational problems associated with martensitic branching

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

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

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

a mesh transformation method for computing microstructures a numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition numerical examples are given to show that the method especially when applied together with a continuation method and some other numerical techniques is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

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

microscopic profiles of shock waves and ambiguities in multiplications of distributions a rigorous mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form the evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

ls μ averaging domains abstract in this paper we first introduce l s averaging domains which are generalizations of l s averaging domains introduced by s g staples we characterize l s averaging domains using the quasihyperbolic metric as applications we prove norm inequalities for conjugate a harmonic tensors in l s averaging domains which can be considered as generalizations of the hardy and littlewood theorem for conjugate harmonic functions finally we give applications to quasiconformal and quasiregular mappings

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

a trace preserving operator and applications abstract we construct a trace preserving operator which improves the integrability of functions in sobolev classes refining the ones available in literature as applications we prove a c 1 partial regularity result for local minimizers of quasiconvex integral functionals satisfying non standard p q growth conditions in the borderline case p n 1 and q n and a global integrability result for weak solutions to a nonlinear elliptic system

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

determination of optimal data placement for psychometric function estimation a computer simulation psychometric functions are used to relate the responses of a subject to physical stimuli in a variety of psychophysical tasks however it is time consuming to obtain data to determine a psychometric function if many stimulus levels and many trials are required a computer simulation was conducted to determine the minimum number of data points needed for such a determination the computer simulation also determined the optimal placements of the stimuli and the number of trials per datum point for psychometric function determinations results indicate that a 2 point sampling method with 30 50 trials per point at optimal locations can produce a psychometric function with accurate spread and threshold estimates in a yes no paradigm however the 4 point sampling method yields statistically smaller variances of the estimates for the 2 alternative forced choice paradigm at least 120 trials per point are needed for the 2 point sampling method s estimated parameters to differ from the known parameter values by less than 5 the simulation results suggest that 3 alternative or 4 alternative forced choice is preferable to 2 alternative furthermore when a criterion free paradigm is not required the yes no paradigm is a better procedure than m alternative forced choice for obtaining the corresponding psychometric function because of smaller standard deviation of the estimates and smaller number of trials point required

a split godunov scheme for solving one dimensional hyperbolic systems in a nonconservative form in this paper we developed a theoretical study for nonconservative sytems in one dimension in order to construct numerical schemes for solving the riemann problem the nonconservative form of our model system required the use of a well adapted theory in order to give us a sense of our problem we chose a framework of generalized functions for solving a scalar hyperbolic equation with a discontinuous coefficient sigma t u sigma x approx 0 where u is the velocity solution of a burgers s equation after an explicit solution of the riemann problem we derived godunov split schemes for computing an approximate solution of the cauchy problem we applied our study to a system modeling elasticity and a system modeling gas dynamics some stability properties of a scheme and its convergence to a generalized solution are proved for the first model numerical experiments confirmed this convergence result for the second model calculations of flows containing weak to moderate shocks showed that conservation errors are reduced when the mesh is refined but were not entirely eliminated

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

weighted norm inequalities for conjugatea harmonic tensors abstract we prove weighted normal inequalities for conjugate a harmonic tensors in john domains which can be considered as generalizations of the hardy and littlewood theorem for conjugate harmonic functions

some examples of rank one convex functions in dimension two keywords sobolev space quasiconvexity rank one convexity reference caa article 1990 004 record created on 2008 11 25 modified on 2017 05 12

a variational view of partial brittle damage evolution under time dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state elaborating on prior work 14 we establish the existence of a relaxed variational evolution where at each time the two states of the material combine to form a fine mixture optimal from the standpoint of the applied load at that time yet preserving the irreversibility of the damaging process

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

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

heterogeneous thin films of martensitic materials we study the effective behavior of heterogeneous thin films with three competing length scales the film thickness and the length scales of heterogeneity and material microstructure we start with three dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der waals type and derive the effective energy density as all length scales tend to zero with given limiting ratios we do not require any a priori selection of asymptotic expansion or ansatz in deriving our results depending on the dominating length scale the effective energy density can be identified by three procedures averaging homogenization and thin film limit we apply our theory to martensitic materials with multi well energy density and use a model example to show that the shape memory behavior can crucially depend on the ratios of these length scales we comment on the effective conductivity of linear composites and also on multilayers made of shape memory and elastic materials

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

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

viscous shock profiles and primitive formulations weak solutions of hyperbolic systems in primitive nonconservative form for which a consistent conservation form exists are considered it is shown that for primitive formulations shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two consistent viscous shock profiles are enforced by adding scheme dependent small viscous perturbations that account for leading order conservation errors the resulting primitive algorithm is conservative to the order of the approximation one dimensional euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms while not eliminating conservation errors entirely it is found that for a wide range of problems both conservative and primitive flow calculations are of comparable quality

on nonconservative algorithms for grid interfaces in computations of fluid flows by domain decomposition methods the necessity of conservation at grid interfaces is now widely claimed in this paper we consider nonconservative algorithms for the interfaces our investigation begins with discussions about typical nonconservative interface treatments for one dimensional calculations the analysis shows that the conservation error of a numerical solution caused by a nonconservative interface matching has an upper bound when the solution itself is bounded furthermore if the numerical solution converges as the mesh size goes to zero it converges to a weak solution of the problem under certain conditions that may be detected numerically also we acquire similar results for two dimensional calculations on grids intersecting with each other in an arbitrary way in order to illustrate the theoretical results we present numerical examples and demonstrate that under those conditions conservation error reduces and accuracy for jumps as well as locations of discontinuities improves as the mesh size decreases

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

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

two weight imbedding inequalities for solutions to the a harmonic equation abstract we obtain the two weight imbedding inequalities for solutions to the a harmonic equation and explore their applications to the k quasiregular mappings in r n these results can be used to study the properties of the homotopy operator t l p d l w 1 p d l 1 l 1 2 n and to establish the weighted l p estimate for solutions to the a harmonic equation

a maximum likelihood method for estimating thresholds in a yes no task a maximum likelihood procedure for estimating threshold values in a yes no task is presented in computer simulations of this procedure it is demonstrated that the variability of the threshold estimates is little affected by the density of the hypotheses tested for a fixed range or by serious misestimates of the slope of the psychometric functions the threshold value is also largely independent of the starting value of the signal the standard deviation of the threshold estimates appears to decrease with the square root of the number of trials with a 2 to 3 db standard deviation possible if only 12 trials are used in the threshold estimates data are presented using human listeners tested on 5 days two threshold estimates based on 12 trials were made at each of the six audiometric frequencies on each day the mean data appear sensible and the standard deviation of the measured thresholds is about 3 db using this procedure it takes less than 3 min to measure the audiogram for a single ear

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

an integral representation theorem for lower semicontinuous envelopes of integral functionals in this paper an integral representation theorem on sequentially weakly lower semicontinuous envelope of multiple integral functionals is proved for integrands which satisfy growth conditions of order p and have lower compactness property the result generalizes the standard results in the area

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

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

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

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

short note a comment on the computation of non conservative products we are interested in the solution of non conservative hyperbolic systems and consider in particular the so called path conservative schemes see e g 2 3 which rely on the theoretical work in 1 the example of the standard euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach

characterization of generalized gradient young measures generated by sequences in w1 1 and bv generalized young measures as introduced by diperna and majda commun math phys 108 667 689 1987 provide a quantitative tool for studying the one point statistics of oscillation and concentration in sequences of functions in this work after developing a functional analytic framework for such measures including a compactness theorem and results on the generation of such young measures by l1 bounded sequences or even by sequences of bounded radon measures we turn to investigation of those young measures that are generated by bounded sequences of w1 1 gradients or bv derivatives we provide several techniques to manipulate such measures including shifting averaging and approximation by piecewise homogeneous young measures and then establish the main new result of this work the duality characterization of the set of bv or w1 1 gradient young measures in terms of jensen type inequalities for quasiconvex functions with linear growth at infinity this result is the natural generalization of the kinderlehrer pedregal theorem arch ration mech anal 115 329 365 1991 j geom anal 4 59 90 1994 for classical young measures to the w1 1 and bv case and contains its version for weakly converging sequences in w1 1 as a special case finally we give an application to a new lower semicontinuity theorem in bv

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

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 high resolution godunov method for compressible multi material flow on overlapping grids a numerical method is described for inviscid compressible multi material flow in two space dimensions the flow is governed by the multi material euler equations with a general mixture equation of state composite overlapping grids are used to handle complex flow geometry and block structured adaptive mesh refinement amr is used to locally increase grid resolution near shocks and material interfaces the discretization of the governing equations is based on a high resolution godunov method but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard conservative shock capturing schemes the energy correction is constructed based on a uniform pressure velocity flow and is significant only near the captured interface a variety of two material flows are presented to verify the accuracy of the numerical approach and to illustrate its use these flows assume an equation of state for the mixture based on the jones wilkins lee jwl forms for the components this equation of state includes a mixture of ideal gases as a special case flow problems considered include unsteady one dimensional shock interface collision steady interaction of a planar interface and an oblique shock planar shock interaction with a collection of gas filled cylindrical inhomogeneities and the impulsive motion of the two component mixture in a rigid cylindrical vessel

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

the effective bulk energy of the relaxed energy of multiple integrals below the growth exponent the characterization of the bulk energy density of the relaxation in w ft r of a functional f u fi i f vu dx jn is obtained for p q q n where u w ft r and is a continuous function on the set of d x n matrices verifying o 0 and 1 q oo typical examples may be found in cavitation and related theories standard techniques cannot be used due to the gap between the exponent q of the growth condition and the exponent p of integrability of the macroscopic strain vu a recently introduced global method for relaxation and fine sobolev trace and extension theorems are applied 1991 mathematics subject classification amer math soc 49q20 49j45

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms