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

minimally invasive surgery for ricci flow singularities in this paper we construct smooth forward ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on sn 1 without performing an intervening surgery in the restrictive context of rotational symmetry this construction gives evidence in favor of perelman s hope for a canonically defined ricci flow through singularities

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

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

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

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

a compactness property for solutions of the ricci flow on orbifolds in this paper we consider ricci flow on orbifolds with at most isolated singularities we generalize two results of r hamilton to orbifolds the first result is the compactness property of ricci flow and the second is an injectivity radius estimate for ricci flow in the proof of the compactness property we give an orbifold version of the injectivity radius estimate of j cheeger m gromov and m taylor also along the way we prove a compactness property of riemannian orbifolds

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

on the weak kahler ricci flow in this paper we define and study the kahler ricci flow with initial data not being smooth and discuss some natural applications

the relaxation of two well energies with possibly unequal moduli the elastic energy of a multiphase solid is a function of its microstructure determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect microstructure evolution and optimal design mathematically the problem is to determine the relaxation under fixed phase fraction of a multiwell energy this paper addresses two such problems in the geometrically linear setting first in two dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy with arbitrary elastic moduli and transformation strains and provide a characterization of the optimal microstructures and the associated strain second in three dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy when either 1 both elastic moduli are isotropic or 2 the elastic moduli are well ordered and the smaller elastic modulus is isotropic in both cases we impose no restrictions on the transformation strains we provide a characterization of the optimal microstructures and the associated strain we also compute a lower bound that is optimal except possibly in one regime when either 1 both elastic moduli are cubic or 2 the elastic moduli are well ordered and the smaller elastic modulus is cubic for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal in all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain in both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction we also characterize optimal microstructures under applied stress

super ricci flows for metric measure spaces abstract we introduce the notions of super ricci flows and ricci flows for time dependent families of metric measure spaces x d t m t t i the former property is proven to be stable under suitable space time versions of mgh convergence uniformly bounded families of super ricci flows are compact in the spirit of the synthetic lower ricci bounds of lott sturm villani for static metric measure spaces the defining property for super ricci flows is the dynamic convexity of the boltzmann entropy ent m t regarded as a functions on the time dependent geodesic space p x w t t i for ricci flows in addition a nearly dynamic concavity of the boltzmann entropy is requested alternatively super ricci flows will be studied in the framework of the calculus of bakry emery ledoux and equivalence to gradient estimates will be derived for both notions of super ricci flows also enforced versions involving an upper dimension bound n will be presented

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

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

projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control abstract this paper investigates the projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control the pinning control method is employed in designing controllers for guaranteeing projective cluster synchronization based on the lyapunov stability theory and well known barbalat s lemma some new sufficient conditions are derived to guarantee the projective cluster synchronization of the fractional order complex network a numerical simulation is given to verify the effectiveness of the theoretical results

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

on non negative quasiconvex functions with unbounded zero sets we construct nontrivial non negative quasiconvex functions denned on m 2 2 with p th order growth such that the zero sets of the functions are lipschitz graphs of mappings from subsets of a fixed two dimensional subspace to its orthogonal complement we assume that the graphs do not have rank one connections with the lipschitz constants sufficiently small in particular we are able to construct quasiconvex functions which are homogeneous of degree p p 1 and conjugating invariant

higher order quasiconvexity reduces to quasiconvexity in this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions precisely it is proved that a smooth strictly 2 quasiconvex function with p growth at infinity p 1 is the restriction to symmetric matrices of a 1 quasiconvex function with the same growth as a consequence lower semicontinuity results for second order variational problems are deduced as corollaries of well known first order theorems

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

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

mean curvature flow with surgery of mean convex surfaces in mathbb r 3 we define a notion of mean curvature flow with surgery for two dimensional surfaces in mathbb r 3 with positive mean curvature our construction relies on the earlier work of huisken and sinestrari in the higher dimensional case one of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author invent math 2015

on various semiconvex relaxations of the squared distance function for the euclidean squared distance function f dist 2 k with k m n n we show that k is convex if and only if f equals either its rank one convex quasiconvex or polyconvex relaxations we also establish that if i k is compact and contractible or ii dim c k k n k is convex if and only if f equals one of the semiconvex relaxations when dist 2 p k is sufficiently large and for case i p m nxn for case ii p e k a k dimensional plane containing c k we also give some estimates of the difference between dist 2 p k and its semiconvex relaxations some possible extensions to more general p distance functions are also considered

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

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

young measure solutions and instability of the one dimensional perona malik equation we apply a variational approach to the one dimensional version of the widely used perona malik equation in image processing we rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2 x 2 matrices m 2x2 we then use the solutions of some heat equations as the centre of the mass for the young measure valued solutions to construct the approximate solutions by using simple laminates the approximate solutions can be viewed as solutions of a perturbation problem by w 1 w p or w 1 w infinity functions the sequences of the approximate solutions generates young measure valued solutions our results also show that the solutions of the one dimensional perona malik equation are unstable under small w 1 w infinity perturbations

restrictions on microstructure we consider the following question given a set of matrices with no rank one connections does it support a nontrivial young measure limit of gradients our main results are these a a young measure can be supported on four incompatible matrices b in two space dimensions a young measure cannot be supported on finitely many incompatible elastic wells c in three or more space dimensions a young measure can be supported on three incompatible elastic wells and d if supports a nontrivial young measure with mean value 0 then the linear span of must contain a matrix of rank one

stationary twists and energy minimizers on a space of measure preserving maps abstract let r n be a bounded lipschitz domain f r n n r a suitably quasiconvex integrand and consider the energy functional f u f u over the space of measure preserving maps a p u w 1 p r n u x det u 1 a e in in this paper we discuss the question of existence of multiple strong local minimizers for f over a p moreover motivated by their significance in topology and the study of mapping class groups we consider a class of maps referred to as twists and examine them in connection with the corresponding euler lagrange equations and investigate various qualitative properties of the resulting solutions the stationary twists particular attention is paid to the special case of the so called p dirichlet energy i e when f p 1 p

on the l function and the reduced volume of perelman ii the main purpose of this paper is to present a number of analytic and geometric properties of the l function and the reduced volume of perelman including in particular the monotonicity the upper bound and the rigidities of the reduced volume

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

weak solutions to degenerate complex monge ampere flows ii abstract studying the long term behavior of the kahler ricci flow on mildly singular varieties one is naturally led to study weak solutions of degenerate parabolic complex monge ampere equations the purpose of this article the second of a series on this subject is to develop a viscosity theory for degenerate complex monge ampere flows on compact kahler manifolds our general theory allows in particular to define and study the normalized kahler ricci flow on varieties with canonical singularities generalizing results of song and tian

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

optimal internal stabilization of the linear system of elasticity we address the non linear optimal design problem which consists in finding the best position and shape of a feedback damping mechanism for the stabilization of the linear system of elasticity non existence of classical designs are related to the over damping phenomenon therefore by means of young measures a relaxation of the original problem is proposed due to the vector character of the elasticity system the relaxation is carried out through div curl young measures which let the analysis be direct and the dimension independent finally the relaxed problem is solved numerically and a penalization technique to recover quasi optimal classical designs from the relaxed ones is implemented in several numerical experiments

super ricci flows for weighted graphs abstract we present a notion of super ricci flow for time dependent finite weighted graphs a challenging feature is that these flows typically encounter singularities where the underlying graph structure changes our notion is robust enough to allow the flow to continue past these singularities as a crucial tool for this purpose we study the heat flow on such singular time dependent weighted graphs with changing graph structure we then give several equivalent characterizations of super ricci flows in terms of a discrete dynamic bochner inequality gradient and transport estimates for the heat flow and dynamic convexity of the entropy along discrete optimal transport paths the latter property can be used to show that our notion of super ricci flow is consistent with classical super ricci flows for manifolds or metric measure spaces in a discrete to continuum limit

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

mean curvature flow with surgery we give a new proof for the existence of mean curvature flow with surgery of 2 convex hypersurfaces in r as announced in hk13a our proof works for all n 3 including mean convex surfaces in r we also derive a priori estimates for a more general class of flows in a local and flexible setting

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