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

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

on conformally invariant equations on rn ii exponential invariance abstract in this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on r n which exhibit conformal invariance of exponential type in this way we intend to complete the work that we undertook in li et al 2011 2 where we introduced the family of elementary conformal tensors t m u in order to describe all fully nonlinear differential operators of any integer order on r n which are conformally invariant of degree 0 examples of the differential operators that we study in this paper are those related to the q curvature equation on r 4 and to the gauss equation on r 2

polyharmonic k hessian equations in rn abstract this work is focused on the study of the nonlinear elliptic higher order equation m u s k u f x r n where the k hessian s k u is the kth elementary symmetric polynomial of eigenvalues of the hessian matrix of the solution and the datum f belongs to a suitable functional space this problem is posed in r n and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m n questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed on the way to construct these proofs some technical results such as a fixed point theorem and a refinement of the critical sobolev embedding which could be of independent interest are introduced

a compactness theorem for a fully nonlinear yamabe problem under a lower ricci curvature bound abstract we prove compactness of solutions of a fully nonlinear yamabe problem satisfying a lower ricci curvature bound when the manifold is not conformally diffeomorphic to the standard sphere this allows us to prove the existence of solutions when the associated cone satisfies 1 which includes the k yamabe problem for k not smaller than half of the dimension of the manifold

the k yamabe flow on manifolds with boundary abstract we consider the problem of conformally deforming a metric by a fully nonlinear flow on a compact manifold with totally geodesic boundary we prove that the k curvature defined by the modified schouten tensor always can be deformed in a natural way to a constant at exponential rate and the boundary preserves totally geodesic

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

toric hilbert schemes we introduce and study the toric hilbert scheme that parametrizes all ideals with the same multigraded hilbert function as a given toric ideal

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

prescribing symmetric functions of the eigenvalues of the ricci tensor we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the ricci tensor we prove an existence theorem for a wide class of symmetric functions on manifolds with positive ricci curvature provided the conformal class admits an admissible metric

existence of radial solutions to biharmonic k hessian equations abstract this work presents the construction of the existence theory of radial solutions to the elliptic equation 2 u 1 k s k u f x x b 1 0 r n provided either with dirichlet boundary conditions u n u 0 x b 1 0 or navier boundary conditions u u 0 x b 1 0 where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f l 1 b 1 0 while r we prove the existence of a caratheodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided is small enough moreover we prove that the solvability set of is finite giving an explicity bound of the extreme value

a fully nonlinear conformal flow on locally conformally flat manifolds we study a fully nonlinear flow for conformal metrics the long time existence and the sequential convergence of flow will be established for locally conformally flat manifolds as an application we solve the k yamabe problem for locally conformal flat manifolds when k n 2

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

existence for a k hessian equation involving supercritical growth abstract in this paper we use variational techniques to give existence results for the problem s k u f x u in u 0 in u 0 on where s k u is the k hessian operator and f x u is a supercritical nonlinearity in the sense introduced by k tso ann inst henri poincare 1990 using some ideas from a celebrated article by brezis and nirenberg we show existence of a positive solution considering supercritical nonlinearities which is surprising given the validity of the pohozaev identity

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

on conformally invariant equations on rn abstract in this paper we provide a complete characterization of fully nonlinear conformally invariant differential operators of any integer order on r n which extends the result proved for operators of the second order by a li and the first author in li and li 2003 1 in particular we prove existence and uniqueness of a family of tensors suitably invariant under mobius transformations which are the basic building blocks that appear in the definition of all conformally invariant differential operators on r n we also explicitly compute the tensors that are related to operators of order up to four

the yamabe problem for higher order curvatures let m be a compact riemannian manifold of dimension n 2 the k curvature for k 1 2 n is defined as the k th elementary symmetric polynomial of the eigenvalues of the schouten tenser the k yamabe problem is to prove the existence of a conformal metric whose k curvature is a constant when k 1 it reduces to the well known yamabe problem under the assumption that the metric is admissible the existence of solutions is known for the case k 2 n 4 for locally conformally flat manifolds and for the cases k n 2 in this paper we prove the solvability of the k yamabe problem in the remaining cases k n 2 under the hypothesis that the problem is variational this includes all of the cases k 2 as well as the locally conformally flat case

the constraint equations of lovelock gravity theories a new σk yamabe problem this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock theories are close to general relativity this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock t

complete moduli in the presence of semiabelian group action i prove the existence and describe the structure of moduli space of pairs p consisting of a projective variety p with semiabelian group action and an ample cartier divisor on it satisfying a few simple conditions every connected component of this moduli space is proper a component containing a projective toric variety is described by a configuration of several polytopes the main one of which is the secondary polytope on the other hand the component containing a principally polarized abelian variety provides a moduli compactification of a g the main irreducible component of this compactification is described by an infinite periodic analog of the secondary polytope and coincides with the toroidal compactification of a g for the second voronoi decomposition

conformal metrics with prescribed curvature functions on manifolds with boundary we study the dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on riemannian manifolds with boundary as a consequence we prove the existence of a conformal metric given its value on the boundary as a prescribed metric conformal to the induced background metric with a prescribed curvature function of the schouten tensor

cyclic concatenated codes with constacyclic outer codes the known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code the outer code is cyclic and the lengths of the inner and outer codes are relatively prime it is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code moreover it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes this description of cyclic codes of composite length leads to the introduction of the concept of a poor code length pcl all but low rate codes of this length have a poor minimum distance a pcl is directly related to the existence of irreducible binomials in the binary case the first few pcls are 9 25 27 45 49 75 81 and 99 arbitrarily long binary cyclic codes that are better than binary bch codes of primitive length are constructed

global regularity of solutions of nonlinear second order elliptic and parabolic differential equations on considere des solutions du probleme de dirichlet pour une equation elliptique totalement non lineaire dans un domaine borne lisse et du probleme parabolique analogue

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

a proof of hessian sobolev inequality in this paper taking the hessian sobolev inequality 0 p k x j wang 1994 2 as the starting point we give a proof of the hessian sobolev inequality when k p k where k is the critical sobolev embedding index of k hessian type we also prove that k is optimal by one dimensional hardy s inequality c 2011 elsevier ltd all rights reserved

hessian measures iii abstract in this paper we continue previous investigations into the theory of hessian measures we extend our weak continuity result to the case of mixed k hessian measures associated with k tuples of k convex functions on domains in euclidean n space k 1 2 n applications are given to capacity quasicontinuity and the dirichlet problem with inhomogeneous terms continuous with respect to capacity or combinations of dirac measures

on polyharmonic regularizations of k hessian equations variational methods abstract this work is devoted to the study of the boundary value problem 1 u 1 k s k u f x r n u n u n 2 u n 1 u 0 x where the k hessian s k u is the k th elementary symmetric polynomial of eigenvalues of the hessian matrix and the datum f obeys suitable summability properties we prove the existence of at least two solutions of which at least one is isolated strictly by means of variational methods we look for the optimal values of n that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results