asymptotic behavior of solutions of a k hessian evolution equation abstract we study the long time behavior of solutions of the k hessian evolution equation u t s k d 2 u posed on a bounded domain of the n dimensional space with homogeneous boundary conditions to this end we construct a separable solution and we show that the long time behavior of u is precisely described by this special solution further we initiate the study of that equation on the entire space providing a new class of explicit and radially symmetric self similar solutions that we call k barenblatt solutions these solutions present some common properties as those of well known barenblatt solutions for the porous media equation and the p laplacian equation

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

a novel adaptive finite element method for the ground state solution of bose einstein condensates abstract in this study we propose a novel adaptive finite element method for the ground state solution of bose einstein condensates bec different from the classical adaptive scheme applied to bec which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low dimensional correction space further the linear elliptic boundary value problem is solved by adaptive multigrid iterations since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces the solving efficiency can be improved evidently in addition the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods this paper introduces a method of constructing nonconforming finite elements which can produce lower bounds for the eigenvalues of elliptic operators based on such nonconforming discrete eigenfunctions we propose a simple method to produce upper bounds of eigenvalues more precisely we construct conforming approximations of exact eigenfunctions by a projection average interpolation operator of nonconforming discrete eigenfunctions after showing the approximation property of the projection average interpolation operator we prove that the rayleigh quotients of the aforementioned conforming approximations are convergent to the exact eigenvalues from above finally we combine lower and upper bounds of eigenvalues to obtain high accuracy approximations of eigenvalues numerical examples verify our theoretical results

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

a multigrid method for eigenvalue problem a multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem in this scheme solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space besides the multigrid scheme all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems the total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem therefore this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving some numerical experiments are presented to validate the efficiency of the method

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

classification of type i and type ii behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume p ps n 2n 2 and that the initial data is bounded possibly sign changing our first goal is to establish various characterizations of type i and type ii blow ups among many other things we show that the following conditions are equivalent a the blow up is of type ii b the rescaled solution w y s converges to either y or y as s where denotes the singular stationary solution c u x t x tends to 1 as x 0 where t is the blow up time our second goal is to study continuation beyond blow up among other things we show that if a blow up is of type i and incomplete then its limit l1 continuation becomes smooth immediately after blow up and that type i blow up implies type i regularization that is t t 1 p 1 u t l is bounded as t t we also give various criteria for complete and incomplete blow ups

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

a multilevel correction adaptive finite element method for kohn sham equation in this paper an adaptive finite element method is proposed for solving kohn sham equation with the multilevel correction technique in the method the kohn sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes a main feature of the method is that solving large scale kohn sham system is avoided effectively and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method hence the significant acceleration can be obtained on solving kohn sham equation with the proposed multilevel correction technique the performance of the method is examined by a variety of numerical experiments

monge problem in metric measure spaces with riemannian curvature dimension condition abstract we prove the existence of solutions for the monge minimization problem addressed in a metric measure space x d m enjoying the riemannian curvature dimension condition rcd k n with n for the first marginal measure we assume that 0 m as a corollary we obtain that the monge problem and its relaxed version the monge kantorovich problem attain the same minimal value moreover we prove a structure theorem for d cyclically monotone sets neglecting a set of zero m measure they do not contain any branching structures that is they can be written as the disjoint union of the image of a disjoint family of geodesics

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

existence of optimal transport maps for crystalline norms we show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in n assuming that the initial distribution of mass is absolutely continuous with respect to mathcal l n the proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the euclidean distance as cost function moreover improving a construction by larman we show the existence of a nikodym set in 3 having full measure in the unit cube intersecting each element of a family of pairwise disjoint open lines only in one point this example can be used to show that the regularity of the decomposition in transport rays plays an essential role in sudakov type arguments for proving the existence of optimal transport maps

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

superconvergence analysis and error expansion for the wilson nonconforming finite element in this paper the wilson nonconforming finite element is considered for solving a class of two dimensional second order elliptic boundary value problems superconvergence estimates and error expansions are obtained for both uniform and non uniform rectangular meshes a new lower bound of the error shows that the usual error estimates are optimal finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms

the lower bound property of the morley element eigenvalues in this paper we prove that the morley element eigenvalues approximate the exact ones from below on regular meshes including adaptive local refined meshes for the fourth order elliptic eigenvalue problems with the clamped boundary condition in any dimension and we implement the adaptive computation with the morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth order elliptic eigenvalue problem with variable coefficients

two sided bounds for eigenvalues of differential operators with applications to friedrichs poincare trace and similar constants we present a general numerical method for computing guaranteed two sided bounds for principal eigenvalues of symmetric linear elliptic differential operators the approach is based on the galerkin method on the method of a priori a posteriori inequalities and on a complementarity technique the two sided bounds are formulated in a general hilbert space setting and as a byproduct we prove an abstract inequality of friedrichs poincare type the abstract results are then applied to friedrichs poincare and trace inequalities and fully computable two sided bounds on the optimal constants in these inequalities are obtained accuracy of the method is illustrated in numerical examples

nonconforming finite element analysis for poisson eigenvalue problem the main aim of this paper is to study a nonconforming quadrilateral finite element named modified quasi wilson element approximation to poisson eigenvalue problem firstly by employing a special property of this element when u h 3 the consistency error is of order o h 2 which is one order higher than its interpolation error o h and the interpolation postprocessing technique the superclose and superconvergence results of order o h 2 for the exact solution of eigenvector u in broken h 1 norm are deduced on generalized rectangular meshes and rectangular meshes respectively secondly it is proved that the consistency error even can reach o h 4 order for arbitrary quadrilateral meshes when u h 5 this is a new astonishing feature which has never been discovered subsequently based on the above characteristic and some asymptotic expansions of the conforming bilinear finite element the extrapolation solution of order o h 4 for eigenvalue is derived finally some numerical results are provided to verify the theoretical analysis

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159