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

the quermassintegral inequalities for k convex starshaped domains abstract we give a proof of the isoperimetric inequality for quermassintegrals of non convex starshaped domains using a result of gerhardt c gerhardt flow of nonconvex hypersurfaces into spheres j differential geometry 32 1990 299 314 and urbas j urbas on the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures math z 205 1990 355 372 on an expanding geometric curvature flow

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

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

diffusion approximation for slow motion in fully coupled averaging in systems which combine fast and slow motions it is usually impossible to study directly corresponding two scale equations and the averaging principle suggests to approximate the slow motion by averaging in fast variables we consider the averaging setup when both fast and slow motions are diffusion processes depending on each other fully coupled and show that there exists a diffusion process which approximates the slow motion in the l 2 sense much better than the averaged motion prescribed by the averaging principle

semi linear liouville theorems in the heisenberg group via vector field methods we consider a global liouville type theorem for semi linear elliptic equation on heisenberg group our main idea is the method of integration by parts for a vector field analogous to the quantity in gidas and spruck 1981 5 c 2009 elsevier inc all rights reserved

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

on a conformally invariant variational problem on differential forms abstract we consider a conformally invariant constrained variational problem on p differential forms for p dim m 2 2 coming from conformal geometry and the associated yamabe type system of partial differential equations it is shown that the infimum p m of the functional is not bigger than p r n and if p m p r n then p m is achieved by a c 2 p form for some 0

the k yamabe solitons and the quotient yamabe solitons abstract in this paper we consider the solitons to the fully nonlinear type yamabe flows we prove that any k yamabe soliton with positive k curvature must have constant k curvature on closed locally conformally flat manifolds we also prove that any quotient yamabe soliton with positive k and l curvatures must have constant k l curvature on closed locally conformally flat manifolds similar results are also proved on compact locally conformally flat manifolds with totally geodesic boundary

some implications of renormalization group theoretical ideas to statistics renormalization group rg theory and statistical data analysis are obviously closely related reductive rg can provide a natural framework to understand asymptotic estimate problems some observations relevant to this problem and an example of a data mining algorithm inspired by the consideration are exhibited

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

inequalities for quermassintegrals on k convex domains abstract in this paper we study the aleksandrov fenchel inequalities for quermassintegrals on a class of non convex domains our proof uses optimal transport maps as a tool to relate curvature quantities of different orders defined on the boundary of the domain

cr yamabe constant cr yamabe flow and its soliton abstract on a compact strictly pseudoconvex cr manifold m we consider the cr yamabe constant of its infinite conformal covering by using the maximum principles we then prove a uniqueness theorem for the cr yamabe flow on a complete noncompact cr manifold finally we obtain some properties of the cr yamabe soliton on complete noncompact cr manifolds

isoperimetric type problems and alexandrov fenchel type inequalities in the hyperbolic space abstract in this paper we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space h n by using quermassintegral preserving curvature flows as a byproduct we obtain hyperbolic alexandrov fenchel inequalities

renormalization group and singular perturbations multiple scales boundary layers and reductive perturbation theory perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis with numerous examples we illustrate its application to ordinary differential equation problems involving multiple scales boundary layers with technically difficult asymptotic matching and wkb analysis in contrast to conventional methods the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally although the latter can be reproduced if desired by appropriate expansion of the renormalization group approximant we show that the renormalization group equation may be interpreted as an amplitude equation and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points deriving both amplitude equations and the center manifold s1063651x 96 00506 5

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

the sobolev inequality for complex hessian equations in this paper we study the complex hessian equations by an gradient flow method we prove a sobolev inequality for k plurisubharmonic functions analogous to that for real hessian equations wang in indiana univ math j 43 25 54 1994 lecture notes in mathematics vol 1977 2009

renormalization group theoretical reduction it has been known for some time that singular perturbation and reductive perturbation can be unified from the renormalization group rg theoretical point of view however renormalization group approaches to singularly perturbed problems require explicit perturbation results so they could be complicated practically the approach proposed in this paper has considerably eliminated the need for explicit perturbation results making the rg approach simpler than many conventional singular perturbation approaches consequently we may assert that reductive extraction of global features of the problem is the essence of singular perturbation methods

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

floquet operators without singular continuous spectrum let u be a unitary operator defined on a infinite dimensional separable complex hilbert space h assume there exists a self adjoint operator a on h such that u au a ci k for some positive constant c and compact operator k then assuming the commutators u au a and a u au admit a bounded extension over h we prove the spectrum of the operator u has no singular continuous component and only a finite number of eigenvalues of finite multiplicity we give a localized version of this result and apply it to study the spectrum of the floquet operator of periodic time dependent kicked quantum systems

existence of extremal functions for the stein weiss inequalities on the heisenberg group abstract in this paper we establish the existence of extremals for two kinds of stein weiss inequalities on the heisenberg group more precisely we prove the existence of extremals for the stein weiss inequalities with full weights in theorem 1 1 and the stein weiss inequalities with horizontal weights in theorem 1 4 different from the proof of the analogous inequality in euclidean spaces given by lieb 26 using riesz rearrangement inequality which is not available on the heisenberg group we employ the concentration compactness principle to obtain the existence of the maximizers on the heisenberg group our result is also new even in the euclidean case because we don t assume that the exponents of the double weights in the stein weiss inequality 1 1 are both nonnegative see theorem 1 3 and more generally theorem 1 5 therefore we extend lieb s celebrated result of the existence of extremal functions of the stein weiss inequality in the euclidean space to the case where the exponents are not necessarily both nonnegative see theorem 1 3 furthermore since the absence of translation invariance of the stein weiss inequalities additional difficulty presents and one cannot simply follow the same line of lions idea to obtain our desired result our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities theorem 1 5

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

stationary scattering theory for unitary operators with an application to quantum walks abstract we present a general account on the stationary scattering theory for unitary operators in a two hilbert spaces setting for unitary operators u 0 u in hilbert spaces h 0 h and an identification operator j h 0 h we give the definitions and collect properties of the stationary wave operators the strong wave operators the scattering operator and the scattering matrix for the triple u u 0 j in particular we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide and we derive representation formulas for the stationary wave operators and the scattering matrix as an application we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature

harmonic mean curvature flow and geometric inequalities abstract we employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove alexandrov fenchel type inequalities relating quermassintegrals to the total curvature which is the integral of gaussian curvature on the hypersurface the resulting inequality allows us to use the inverse mean curvature flow to prove alexandrov fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces finally we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h convex hypersurfaces in hyperbolic space

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

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

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 dirichlet problem for the kohn laplacian on the heisenberg group ii for x y t rn rn r denote xj xj 2yj t yj yj 2xj t and l 14 j 1nxj2 yj2 t when n 2q la represents the action of the kohn laplacian b on q forms on the heisenberg group for n n we construct a parametrix for the dirichlet problem in smooth domains d near non characteristic points of d a point w of d is non characteristic if one of x1 xn y1 yn is transverse to d at w this yields sharp local estimates in the dirichlet problem in the appropriate non isotropic lipschitz classes the main new tool is a convolution calculus of pseudo differential operators that can be applied to the relevant layer potentials for which the usual asymptotic composition formula is false characteristic points are treated in part ii

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

the principal eigenvalue of a conformally invariant differential operator with an application to semilinear elliptic pde in this paper we give conditions under which the paneitz operator a conformally invariant fourth order differential operator is positive we also prove a sharp estimate for a related conformal invariant these results are combined to study the existence of a semilinear pde with critical sobolev growth

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

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

renormalization group theoretical reduction of the swift hohenberg model the swift hohenberg model of the cellular pattern formation is exploited with a proto renormalization group rg scheme the method dispenses with the explicit perturbation solutions which are required in the standard rg approach the rg equations obtained are the well known reductive perturbation results such as a rotationally covariant amplitude equation and the nonlinear phase equations

boundary operators associated to the σk curvature abstract we study conformal deformation problems on manifolds with boundary which include prescribing k 0 in the interior in particular we prove a dirichlet principle when the induced metric on the boundary is fixed and an obata type theorem on the upper hemisphere we introduce some conformally covariant multilinear operators as a key technical tool

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

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

exact multilocal renormalization of the effective action application to the random sine gordon model statics and nonequilibrium dynamics we extend the exact multilocal renormalization group rg method to study the flow of the effective action functional this important physical quantity satisfies an exact rg equation which is then expanded in multilocal components integrating the nonlocal parts yields a closed exact rg equation for the local part to a given order in the local part the method is illustrated on the o n model by straightforwardly recovering the ensuremath eta exponent and scaling functions then it is applied to study the glass phase of the cardy ostlund random phase sine gordon model near the glass transition temperature the static correlations and equilibrium dynamical exponent z are recovered and several results are obtained such as the equilibrium two point scaling functions the nonequilibrium finite momentum two time t t ensuremath response and correlations are computed they are shown to exhibit scaling forms characterized by exponents ensuremath lambda r ensuremath ne ensuremath lambda c as well as universal scaling functions that we compute the fluctuation dissipation ratio is found to be nontrivial and of the form x q z t ensuremath t ensuremath t t ensuremath analogies and differences with pure critical models are discussed

conformally invariant monge ampere equations global solutions in this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group so n 1 1 these equations are elliptic and variational using this structure and the conformal invariance we will prove a global uniqueness theorem for solutions in rn with a quadratic growth condition at infinity