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 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

volume comparison and the σk yamabe problem abstract in this paper we study the problem of finding a conformal metric with the property that the k th elementary symmetric polynomial of the eigenvalues of its weyl schouten tensor is constant a new conformal invariant involving maximal volumes is defined and this invariant is then used in several cases to prove existence of a solution and compactness of the space of solutions provided the conformal class admits an admissible metric in particular the problem is completely solved in dimension four and in dimension three if the manifold is not simply connected

exact bounds for the static response set of structures with uncertain but bounded parameters abstract the numerical estimation of the static displacement bounds of structures with uncertain but bounded parameters is considered in this paper by representing each uncertain but bounded parameter as an interval number or vector a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations in which the coefficient matrix and the right hand side term are respectively the interval matrix and the interval vector in this study we present two new simple mathematical proofs of the vertex solution theorem using cramer s rule for solving linear interval equations different from the other proof methods to find the upper and lower bounds on the set of solutions the first takes advantage of optimization theory while the second is based on interval extension by means of a typical example considered first by hansen it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the oettli prager criterion examples of a three stepped beam and a 10 bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method

a new variational characterization of n dimensional space forms a riemannian manifold m n g is associated with a schouten 0 2 tensor cg which is a naturally defined codazzi tensor in case m n g is a locally conformally flat riemannian manifold in this paper we study the riemannian functional f k g f m k c g dvol g defined on m 1 g e m vol g 1 where m is the space of smooth riemannian metrics on a compact smooth manifold m and k c g 1 5 and a conformally flat metric g is a critical point of f 2 m 1 with f 2 g 0 then g must have constant sectional curvature this is a generalization of gursky and viaclovsky s very recent theorem that the critical point of f 2 m 1 with f 2 g 0 characterized the three dimensional space forms

information rate of some classes of non regular languages an automata theoretic approach abstract we show that the information rate of the language accepted by a reversal bounded deterministic counter machine is computable for the nondeterministic case we provide computable upper bounds for the class of languages accepted by multi tape deterministic finite automata the information rate is computable as well

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

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

some properties of the schouten tensor and applications to conformal geometry the riemannian curvature tensor decomposes into a conformally invariant part the weyl tensor and a non conformally invariant part the schouten tensor a study of the kth elementary symmetric function of the eigenvalues of the schouten tensor was initiated in an earlier paper by the second author and a natural condition to impose is that the eigenvalues of the schouten tensor are in a certain cone k we prove that this eigenvalue condition for k n 2 implies that the ricci curvature is positive we then consider some applications to the locally conformally flat case in particular to extremal metrics of k curvature functionals and conformal quermassintegral inequalities using the results of the first and third authors

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

a conformally invariant gap theorem characterizing mathbb cp 2 via the ricci flow we extend the sphere theorem of chang et al publ math inst ht etudes sci 98 105 434 2003 to give a conformally invariant characterization of mathbb cp 2 g fs in particular we introduce a conformal invariant beta m 4 g ge 0 defined on conformal four manifolds satisfying a positivity condition it follows from chang et al 2003 that if 0 le beta m 4 g 0 and 4 le beta m 4 g 0 small enough then m 4 is diffeomorphic to mathbb cp 2 the ricci flow is used in a crucial way to pass from the bounds on beta to pointwise curvature information

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

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 solution of linear interval equations by a linear programming method abstract a method is described for solving a system of n linear equations in n unknowns when the coefficients and constants of the equations are specified as intervals here a solution is considered to be the supplying of optimal interval bounds for the unknowns even for small n such systems of interval equations can be difficult to solve requiring extensive computation the method described is a generalization of a linear programming method proposed long ago by w oettli and leads to a relatively simple solution algorithm applicable to systems of small size

conformal deformations of the smallest eigenvalue of the ricci tensor we consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the ricci tensor is a constant it is related to the notion of minimal volumes in comparison geometry such a metric with the smallest eigenvalue of the ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class the problem is reduced to solve a pucci type equation with respect to the schouten tensor we establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations combining it with the theory of fully nonlinear equations we establish the existence of solutions for this equation

metrics of negative ricci curvature one of the most natural and important topics in riemannian geometry is the relation between curvature and global structure of the underlying manifold for instance complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic furthermore it seemed to be a natural principle that a closed manifold cannot carry two metrics of different signed curvatures as it is a basic fact that this is true for sectional curvature but it turned out to be wrong much later and from a strongly analytic argument for the scalar curvature s since each manifold m n 3 admits a complete metric with s 1 cf aubin a and bland kalka bik hence the situation for ricci curvature ric lying between sectional and scalar curvature seemed to be quite delicate up to now the most general results concerning ric 0 were proved by gao yau gy and brooks br using thurston s theory of hyperbolic threemanifolds viz each closed three manifold admits a metric with ric 0 this is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities gao and yau resp brooks smoothed these singularities to get a regular metric with ric 0 these methods extend to three manifolds of finite type and certain hyperbolic orbifolds in any case the arguments rely on exploiting some extraordinary metric structures whose existence is neither obvious nor conceptually related to the ricci curvature problem indeed the existence depends on the assumption that the manifold is three dimensional and compact moreover this approach does not provide insight into the typical behaviour of metrics with ric 0 since one is led to very special metrics in this article we approach negative ricci curvature using a completely different and new concept which will become even more significant in l2 as we deliberately produce ric 0 actually we will prove the following results in these notes ric g resp r g denotes the ricci tensor resp curvature of a smooth metric g

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

a new conformal invariant on 3 dimensional manifolds abstract by improving the analysis developed in the study of k yamabe problem we prove in this paper the following new inequality for any 3 dimensional riemannian manifold m 3 g of non negative scalar curvature m 1 g d v g 2 3 vol g m 2 g d v g equality holds if and only if m 3 g is a space form