a local proof for the characterization of young measures generated by sequences in bv this work presents a new proof of the recent characterization theorem for generalized young measures generated by sequences in bv by kristensen and rindler 2010 14 the present argument is based on a localization technique together with a local hahn banach argument in novel function spaces combined with an application of alberti s rank one theorem this strategy avoids employing a relaxation theorem as in the previously known proof and the new tools introduced in its course should prove useful in other contexts as well in particular we introduce homogeneous young measures separately at regular and singular points which exhibit rather different behavior than the classical homogeneous young measures as an application we show how for bv young measures with an atomic part one can find a generating sequence respecting this structure

rank one property for derivatives of functions with bounded variation when u is a function of bounded variation of the open set r into r du is a measure on which takes values in the space rm n of all m n matrices such that du i j ui xj for every i j with 1 i m 1 j n and we denote by dsu the part of the measure du which is singular with respect to lebesgue measure we recall that the rank of a matrix a rm n is dimension of the space spanned by its columns in r or by its rows in r and in particular the rank of a is one if and only if there exist e r and r with e 6 0 6 0 so that a e i e ai j ei j for all i j cf remark 1 7 when is a measure which takes values in the space rm n and d d denotes the radon nikodym derivative of with respect to the total variation we say that has rank one when the rank of the matrix d d x is one for almost every x in this paper we prove we prove the following statement see theorem 4 5 corollary 4 6 and remark 4 9

on rank one convex functions that are homogeneous of degree one we show that positively 1 homogeneous rank one convex functions are convex at 0 and at matrices of rank one the result is a special case of an abstract convexity result that we establish for positively 1 homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space from these results we derive a number of consequences including various generalizations of the ornstein l1 non inequalities most of the results were announced in c r acad sci paris ser i 349 407 409 2011

metric currents and alberti representations we relate ambrosio kirchheim metric currents to alberti representations and weaver derivations in particular given a metric current t we show that if the module span class formulatext stixsupport mathimg data mathurl science ob mathurl method retrieve eid 1 s2 0 s0022123616302324 mathid si1 gif user 111111111 pii s0022123616302324 rdoc 1 issn 00221236m this extends previous results of williams applications of this theory include an approximation of 1 dimensional metric currents in terms of normal currents and the construction of alberti representations in the directions of vector fields

fine properties of functions with bounded deformation the paper is concerned with the fine properties of functions in the space of functions with bounded deformation we analyse the set of lebesgue points and the set where these functions have one sided approximate limits moreover following the analogy with we decompose the symmetric distributional derivative into an absolutely continuous part a jump part and a cantor part the main result of the paper is a structure theorem for functions showing that these parts of the derivative can be recovered from the corresponding ones of the one dimensional sections moreover we prove that functions are approximately differentiable in almost every point of their domain

dislocation microstructures and the effective behavior of single crystals we consider single crystal plasticity in the limiting case of infinite latent hardening which signifies that the crystal must deform in single slip at all material points this requirement introduces a nonconvex constraint and thereby induces the formation of fine scale structures we restrict attention throughout to linearized kinematics and deformation theory of plasticity which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well defined variational structure analogous to elasticity we first study a scale invariant local problem we show that by developing microstructures in the form of sequential laminates of finite depth crystals can beat the single slip constraint i e the macroscopic relaxed constitutive behavior is indistinguishable from multislip ideal plasticity in a second step we include dislocation line energies and hence a length scale into the model different regimes lead to several possible types of microstructure patterns we present constructions which achieve the various optimal scaling laws and discuss the relation with experimentally known scalings such as the hall petch law

on the structure of a free measures and applications we establish a general structure theorem for the singular part of a free radon measures where a is a linear pde operator by applying the theorem to suitably chosen differential operators a we obtain a simple proof of alberti s rank one theorem and for the first time its extensions to functions of bounded deformation bd we also prove a structure theorem for the singular part of a finite family of normal currents the latter result implies that the rademacher theorem on the differentiability of lipschitz functions can hold only for absolutely continuous measures and that every top dimensional ambrosio kirchheim metric current in rd is a federer fleming flat chain

a global method for relaxation a new method for the identification of the integral representation of a class of functionals defined on bv omega bbb r d times cal a omega where cal a omega represents the family of open subsets of is presented applications are derived such as the integral representation of the relaxed energy in bv omega bbb r d corresponding to a functional defined in w 1 1 omega bbb r d with a discontinuous integrand with linear growth relaxation and homogenization results in sbv omega bbb r d are recovered in the case where bulk and surface energies are present

transport maps non branching sets of geodesics and measure rigidity abstract in this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non branching introduced by rajala sturm 2014 27 in particular it is shown that the qualitative non degeneracy condition introduced by cavalletti huesmann 2015 6 implies that any essentially non branching metric measure space has a unique transport maps whenever the initial measure is absolutely continuous this generalizes a recently obtained result by cavalletti mondino 2017 8 on essentially non branching spaces with the measure contraction condition mcp k n in the end we prove a measure rigidity result showing that any two essentially non branching qualitatively non degenerate measures on a fixed metric spaces must be mutually absolutely continuous this result was obtained under stronger conditions by cavalletti mondino 2016 7 it applies in particular to metric measure spaces with generalized finite dimensional ricci curvature bounded from below

a rademacher type theorem on l2 wasserstein spaces over closed riemannian manifolds abstract let p be any borel probability measure on the l 2 wasserstein space p 2 m w 2 over a closed riemannian manifold m we consider the dirichlet form e induced by p and by the wasserstein gradient on p 2 m under natural assumptions on p we show that w 2 lipschitz functions on p 2 m are contained in the dirichlet space d e and that w 2 is dominated by the intrinsic metric induced by e we illustrate our results by giving several detailed examples

piecewise affine approximations for functions of bounded variation bv functions cannot be approximated well by piecewise constant functions but this work will show that a good approximation is still possible with countably piecewise affine functions in particular this approximation is area strictly close to the original function and the mathrm l 1 l1 difference between the traces of the original and approximating functions on a substantial part of the mesh can be made arbitrarily small necessarily the mesh needs to be adapted to the singularities of the bv function to be approximated and consequently the proof is based on a blow up argument together with explicit constructions of the mesh in the case of mathrm w 1 1 w1 1 sobolev functions we establish an optimal mathrm w 1 1 w1 1 error estimate for approximation by piecewise affine functions on uniform regular triangulations the piecewise affine functions are standard quasi interpolants obtained by mollification and lagrange interpolation on the nodes of triangulations and the main new contribution here compared to for instance clement rairo anal numer 9 r 2 77 84 1975 and verfurth m2an math model numer anal 33 4 1766 1782 1999 is that our error estimates are in the mathrm w 1 1 w1 1 norm rather than merely the mathrm l 1 l1 norm

concentration versus oscillation effects in brittle damage this work is concerned with an asymptotic analysis in the sense of gamma convergence of a sequence of variational models of brittle damage in the context of linearized elasticity the study is performed as the damaged zone concentrates into a set of zero volume and at the same time and to the same order varepsilon the stiffness of the damaged material becomes small three main features make the analysis highly nontrivial at varepsilon fixed minimizing sequences of each brittle damage model oscillate and develop microstructures as varepsilon to 0 concentration and saturation of damage are favoured and the competition of these phenomena translates into a degeneration of the growth of the elastic energy which passes from being quadratic at varepsilon fixed to being of linear growth type in the limit consequently homogenization effects interact with singularity formation in a nontrivial way which requires new methods of analysis in particular the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far we explicitly identify the gamma limit in two and three dimensions for isotropic hooke tensors the expression of the limit effective energy turns out to be of hencky plasticity type we further consider the regime where the divergence remains square integrable in the limit which leads to a tresca type model

characterization of generalized young measures generated by symmetric gradients this work establishes a characterization theorem for generalized young measures generated by symmetric derivatives of functions of bounded deformation bd in the spirit of the classical kinderlehrer pedregal theorem our result places such young measures in duality with symmetric quasiconvex functions with linear growth the local proof strategy combines blow up arguments with the singular structure theorem in bd the analogue of alberti s rank one theorem in bv which was recently proved by the authors as an application of our characterization theorem we show how an atomic part in a bd young measure can be split off in generating sequences

rank one theorem and subgraphs of bv functions in carnot groups abstract we prove a rank one theorem a la g alberti for the derivatives of vector valued maps with bounded variation in a class of carnot groups that includes heisenberg groups h n for n 2 the main tools are properties relating the horizontal derivatives of a real valued function with bounded variation and its subgraph

rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies we extend allard s celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand in particular we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every d dimensional varifold with locally bounded first variation and positive d dimensional density in codimension 1 this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane 2017 wiley periodicals inc

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

directional oscillations concentrations and compensated compactness via microlocal compactness forms this work introduces microlocal compactness forms mcfs as a new tool to study oscillations and concentrations in l p bounded sequences of functions decisively mcfs retain information about the location value distribution and direction of oscillations and concentrations thus extending at the same time the theories of generalized young measures and h measures in l p spaces oscillations and concentrations precisely discriminate between weak and strong compactness and thus mcfs allow one to quantify the difference in compactness the definition of mcfs involves a fourier variable whereby differential constraints on the functions in the sequence can also be investigated easily a distinct advantage over young measure theory furthermore pointwise restrictions are reflected in the mcf as well paving the way for applications to tartar s framework of compensated compactness consequently we establish a new weak to strong compactness theorem in a geometric way after developing several aspects of the abstract theory we consider three applications for lamination microstructures the hierarchy of oscillations is reflected in the mcf the directional information retained in an mcf is harnessed in the relaxation theory for anisotropic integral functionals finally we indicate how the theory pertains to the study of propagation of singularities in certain systems of pdes the proofs combine measure theory young measures and harmonic analysis

on two dimensional percolation we present new series data for both high and low density bond and site percolation on the square lattice the series have been obtained by the finite lattice method and in all cases extend pre existing series an analysis of these series gives refined estimates of critical points critical exponents and amplitudes for bond and site animals and for the percolation probability and mean size exponents

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

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

distributed source localization using esprit algorithm a new algorithm based on esprit is proposed for the estimation of the central angle and angular extension of distributed sources the central angles are estimated using tls esprit for both incoherently distributed id and coherently distributed cd sources for cd sources the extension width is estimated by constructing a one dimensional 1 d distributed source parameter estimator dspe spectrum for each source for id sources the extension widths are estimated using the central moments of the distribution the algorithm can be used for sources with different angular distributions

differentiability of lipschitz functions in lebesgue null sets we show that if n 1 then there exists a lebesgue null set in mathbb r n containing a point of differentiability of each lipschitz function f mathbb r n rightarrow mathbb r n 1 in combination with the work of others this completes the investigation of when the classical rademacher theorem admits a converse avoidance of sigma porous sets arising as irregular points of lipschitz functions plays a key role in the proof

on the two state problem for general differential operators abstract in this note we generalize the ball james rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint in particular we prove the rigidity for approximate solutions to the two state inclusion with incompatible states for merely l 1 bounded sequences in this way our theorem can be seen as a result of compensated compactness in the linear growth setting

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