a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

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

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

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

lower semicontinuity of multiple integrals and the biting lemma weak lower semicontinuity theorems in the sense of chacon s biting lemma are proved for multiple integrals of the calculus of variations a general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions the biting weak limit of the corresponding integrands is characterised via the young measure and related to the weak limit in the sense of measures finally an example is given which shows that the young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case

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

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

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

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

a family of simplex variants solving an m d linear program in expected number of pivot steps depending on d only we present a family of variants of the simplex method which are based on a constraint by constraint procedure the solution to a linear program is obtained by solving a sequence of subproblems with an increasing number of constraints we discuss several probabilistic models for generating linear programs in all of them the underlying distribution is assumed to be invariant under changing the signs of rows or columns in the problem data a weak regularity condition is also assumed under these models for linear programs with d variables and m d inequality constraints the expected number of pivots required by these algorithms is bounded by a function of minm d only in particular this means that for a fixed number of variables the expected number of pivots is bounded by a constant when the number of constraints tends to infinity since smale s original model smale s 1983 on the average speed of the simplex method of linear programming math programming27 satisfies our probabilistic assumptions the same results apply to his model although not to the particular algorithm he analyzes we also present some results for models generating only feasible linear programs and for bland s pivoting rule we conclude with a discussion of our probabilistic models and show why they are inadequate for obtaining meaningful results unless d and m are of the same order of magnitude

newton kantorovitch algorithm applied to an electromagnetic inverse problem an efficient algorithm is presented to compute the shape of perfectly conducting cylinders via knowledge of scattering cross sections this choice of scattering data avoids the difficulties linked with the phase measurements or reconstructions the mathematical analysis is performed in terms of operator and functions and the fundamental instability of the problem is demonstrated then the stability is restored by means of a tikhonov miller regularization the efficiency of the method is outlined by numerical examples references are given which show that the same algorithm applies to other electromagnetic inverse problems especially to gratings

rank one convexity implies quasi convexity on certain hypersurfaces we show that if f m 2 2 r is rank one convex on the hyperboloid is the set of 2 2 real symmetric matrices then f can be approximated by quasi convex functions on m 2 2 uniformly on compact subsets of equivalently every gradient young measure supported on a compact subset of is a laminate

the reconstruction of shape and impedance exploiting space mapping with inverse difference method the reconstruction of shape and impedance obtained from measurements of scattered electromagnetic fields is considered for the two dimensional case space mapping with inverse difference technique which is newly developed to solve inverse problem is used for this purpose one of the advantages of this technique over other space mapping techniques is that it does not need parameter extraction during optimization process as the contour of the cylinder is periodic it can be represented by fourier series measurements of scattered field and fourier coefficients of the contour are used as inputs and outputs respectively the results are compared with exact shapes and good agreement is obtained it is also observed that reconstructions obtained by space mapping based technique are better than those obtained by artificial neural network for exact and noisy data sharp cornered tetragonal contour is also investigated for both noisy and noiseless case in order to show efficiency of the technique

an infeasible exterior point simplex algorithm for assignment problems the so called modified hung rom algorithm based upon theoretical considerations of hirsch paths seems to be one of the most efficient algorithms for assignment problems since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest this paper presents an algorithm of this kind for assignment problems

scarf s procedure for integer programming and a dual simplex algorithm scarf has recently introduced an algorithm for integer programs based on the combinatorial concept of primitive set we show that as the decision variables of the integer program become continuous and the integer program reduces to a linear program the scarf algorithm converges to a dual simplex algorithm for the limit linear programming problem this result is robust in the sense that even before the limit is reached the dual simplex path is contained in the path of primitive sets generated by the scarf algorithm

quasiconvex functions with subquadratic growth we establish the existence of quasiconvex functions which are not convex and which have subquadratic grow that infinity

incompatible sets of gradients and metastability we give a mathematical analysis of a concept of metastability induced by incompatibility the physical setting is a single parent phase just about to undergo transformation to a product phase of lower energy density under certain conditions of incompatibility of the energy wells of this energy density we show that the parent phase is metastable in a strong sense namely it is a local minimizer of the free energy in an l1 neighbourhood of its deformation the reason behind this result is that due to the incompatibility of the energy wells a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus we define and characterize incompatible sets of matrices in terms of which the transition layer estimate at the heart of the proof of metastability is expressed finally we discuss connections with experiments and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis

rank one convexity implies polyconvexity for isotropic objective and isochoric elastic energies in the two dimensional case we show that in the two dimensional case every objective isotropic and isochoric energy function which is rank one convex on gl 2 is already polyconvex on gl 2 thus we negatively answer morrey s conjecture in the subclass of isochoric nonlinear energies since polyconvexity implies quasiconvexity our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular we also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor

smoothness of linear laminates linearly elastic laminates are examples of materials whose equilibrium equations may have only bounded measurable coefficients yet whose solutions may be fairly smooth this is quite different from general experience where regularity of solutions is determined by the closeness of the system to a diagonal one a particular situation where a laminate may appear is a highly twinned elastic or ferroelectric crystal and there are questions related to these materials which make it useful to know some properties of these special systems of equations

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we