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

equations du type de monge ampere sur les varietes riemanniennes compactes ii abstract let vn g be a c compact riemannian manifold without boundary given the following changes of metric g g hess l 2 g g 2 hess where a is a fixed constant we study the corresponding monge ampere equations 1 log g g 1 f p 2 log g g 1 f p we first solve eq 2 under some simple assumptions on f c then using an appropriate change of functions that enables us to take advantage of the estimates just carried out for eq 2 we extend to eq 1 all the results proved in our previous articles 5 6 for the usual monge ampere equation although equation 2 is not locally invertible and does not even admit a solution for all f 0 f c v n a similar change of functions leads to partial results about eq 1 via c2 and c3 estimates for eq 2 eventually we give some comments and errata of our previous article p delanoe j funct anal 41 1981 341 353

nonlinear delay systems controllability abstract this research resolves some controllability questions of general nonlinear delay systems in function space in which earlier solutions of the optimal problem have been shown to exist by using a powerful open mapping theorem and gateaux differentiability we avoid the difficult requirement of frechet differentiability of an operator f wp 1 lp wp 1 which is associated with the nonlinear system properties of the linearized system help ensure the nonlinear system s controllability

normally ordering some multimode exponential operators by virtue of the iwop technique the author shows that the technique of integration within an ordered product iwop provides one with a very simple approach to deriving the normal product form of some multimode exponential operators which greatly simplifies the calculations of normalising some state vectors in hilbert space

on partial s controllability of semilinear partially observable systems in this paper the partial s controllability of semilinear partially observable control systems is investigated the essence of study is the presence of partial observations we define a weakened partial s controllability concept for partially observable systems a sufficient condition for this controllability concept is proved the method of proof differs from the traditional proofs by fixed point theorems the obtained result is demonstrated with examples

a new unitary operator for a pair of coupled oscillators using the iwop integration within ordered product technique a new unitary operator is found that is useful in diagonizing the hamiltonian of a pair of coupled harmonic oscillators the coordinate representation of the unitary operator is presented and is well applied to obtain the wave function of energy eigenstate of the coupled oscillator

equations du type monge ampere sur les varietes riemanniennes compactes i let vn g be a c compact riemannian manifold for a suitable function on vn let us consider the change of metric g g hess and the function as a ratio of two determinants m g g 1 using the method of continuity we first solve in c the problem log m 0 c then under weak hypothesis on f we solve the general equation log m f p f in c vn using a method of iteration our study gives rise to an interesting a priori estimate on which does not occur in the complex case this estimate should enable us to solve the equation above when 0 providing we can overcome difficulties related to the invertibility of the linearised operator this open question will be treated in our next article

global bifurcation and convex solutions for the monge ampere equation abstract we study the following monge ampere equation det d 2 u 1 n f u in u 0 on by bifurcation technique we establish some results about the existence nonexistence uniqueness and multiplicity of convex solutions for this problem our results generalize and improve many important known results from previous literature

ricci iteration on homogeneous spaces the ricci iteration is a discrete analogue of the ricci flow we give the first study of the ricci iteration on a class of riemannian manifolds that are not kahler the ricci iteration in the non kahler setting exhibits new phenomena among them is the existence of so called ancient ricci iterations as we show these are closely related to ancient ricci flows and provide the first nontrivial examples of riemannian metrics to which the ricci operator can be applied infinitely many times in some of the cases we study these ancient ricci iterations emerge in the gromov hausdorff topology from a collapsed einstein metric and converge smoothly to a second einstein metric in the case of compact homogeneous spaces with maximal isotropy we prove a relative compactness result that excludes collapsing our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient ricci flow exists on a given riemannian manifold as well as a method for predicting its limit

permutation operators in hilbert space gained via iwop technique a kind of permutation operators in hilbert space are derived in terms of the technique of integration within ordered product iwop these operators are shown to be quantum maps imaged by certain permutation transformations in classical space some operator identities about the relationship between permutation operators and transposition operators are also derived

canonical coherent state representation of some squeeze operators using the iwop technique integration within an ordered product the authors find canonical coherent state representations of two kinds of squeeze operators these representations manifestly show that they are quantum maps imaged by certain symplectic transformations in coordinate momentum phase space in terms of this formalism the coherent state propagator of parametric amplifiers is easily obtained

normally ordered fermi operator realization of the sun group the mapping of the sun group transformation in grassmann number space into unitary normally ordered and antinormally ordered fermi operator realizations in hilbert space are investigated the unitary fermi operators are evaluated in fermion coherent state representation using the integration within ordered product technique for a fermionic system some new fermi operator identities are thus obtained

rothe s fixed point theorem and controllability of semilinear nonautonomous systems abstract in this paper we apply rothe s fixed point theorem to prove the controllability of the following semilinear system of ordinary differential equations z t a t z t b t u t f t z t u t t 0 z 0 z 0 where z t r n u t r m a t b t are continuous matrices of dimensions n n and n m respectively the control function u belongs to l 2 l 2 0 r m and the nonlinear function f 0 r n r m r n is continuous and there are a b c r and 1 2 1 such that f t z u r n a z r n b u r m c u r m z r n under this condition we prove the following statement if the linear z t a t z t b t u t is controllable then the semilinear system is also controllable on 0 moreover we could exhibit a control steering the nonlinear system from an initial state z 0 to a final state z 1 at time 0

controllability of semilinear impulsive nonautonomous systems in this paper we apply rothe s type fixed point theorem to prove the controllability of the following semilinear impulsive nonautonomous systems of differential equations where a t b t are continuous matrices of dimension n n and n m respectively the control function u belongs to and k 1 2 3 p under additional conditions we prove the following statement if the linear is controllable on 0 then the semilinear impulsive system is also controllable on 0 moreover we could exhibit a control steering the nonlinear system from an initial state z0 to a final state z1 at time 0

metrics with prescribed ricci curvature near the boundary of a manifold suppose m is a manifold with boundary choose a point o in partial m we investigate the prescribed ricci curvature equation mathop mathrm ric nolimits g t in a neighborhood of o under natural boundary conditions the unknown g here is a riemannian metric the letter t on the right hand side denotes a 0 2 tensor our main theorems address the questions of the existence and the uniqueness of solutions we explain among other things how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus mathcal t the paper concludes with a brief discussion of the einstein equation on mathcal t

local solvability of elliptic and curvature equations on compact manifolds we discuss the search of local constraints for a nonlinear elliptic operator on a compact manifold prove a new version of fredholm theorem for a class of operators with vanishing index and apply it to the local solbability of some prescribed curvature equations

scalar curvature functions in a conformal class of metrics and conformal transformations on etudie le probleme de prescrire la courbure scalaire dans une classe conforme pour l action du groupe conforme on montre l universalite des conditions d integrabilite dues a j l kazdan et f w warner