necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

hausdorff dimension of the set of singular pairs in this paper we show that the hausdorff dimension of the set of singular pairs is 4 3 we also show that the action of diag e t e t e 2t on sl 3 r sl 3 z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates answering a question of a n starkov as a by product of the analysis we obtain a higher dimensional generalization of the basic inequalities satisfied by convergents of continued fractions as an illustration of the technique used to compute hausdorff dimension we reprove a result of i j good asserting that the hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2

the heat equation on a cr manifold on considere le souslaplacien b sur une variete compacte cr on developpe un calcul pseudodifferentiel approprie

estimates for sums of eigenvalues of the laplacian the aim of this paper is to give bounds for the eigenvalues of the laplacian on a domain in euclidean space and on a compact riemannian manifold first we consider the eigenvalue problem for the laplacian on a bounded domain in euclidean space under dirichlet and neumann boundary conditions our method for obtaining an upper bound for sums of eigenvalues under dirichlet boundary conditions is closely related to the method used earlier j funct anal 106 1992 353 357 for the task of getting an upper bound for sums of eigenvalues under neumann boundary conditions on the other hand we modify the method used by p li and s t yau comm math phys 88 1983 309 318 for obtaining a lower bound for sums of eigenvalues under dirichlet boundary conditions in order to get a lower bound for sums of eigenvalues under neumann boundary conditions under the assumption that the domain under consideration is lipschitz equivalent to a ball finally we derive estimates for sums of squares of eigenvalues on a compact riemannian manifold without boundary

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

existence of multiple positive solutions to nonhomogeneous schrodinger poisson system in this paper we consider the existence of multiple solutions to the following nonhomogeneous generalized schrodinger poisson system u ku q f u g u h x in r 3 2 qf u in r 3 where q 0 is a parameter 0 h x h x l 2 r 3 and g is asymptotically linear or superliner at infinity we show that there exists q 0 0 such that the system has at least two positive radial solutions for q 0 q 0

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

multiple solutions for a nonhomogeneous schrodinger maxwell system in r 3 abstract the paper considers the following nonhomogeneous schrodinger maxwell system sm u u x u u p 1 u g x x r 3 u 2 x r 3 where 0 p 1 5 and 0 g x g x l 2 r 3 there seem to be no results on the existence of multiple solutions to problem sm for p 1 3 in this paper we find that there is a constant c p 0 such that problem sm has at least two solutions for all p 1 5 provided that g l 2 c p however for p 1 2 we need 0 is small moreover c p p 1 2 p p 1 s p 1 2 p 1 p 1 where s is the sobolev constant

decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

regularity for quasilinear second order subelliptic equations in this paper we study the regularity of solutions of the quasilinear equation where x x1 xm is a system of real smooth vector fields aij b c rm 1 assume that x satisfies the hormander condition and aij x z is positive definite we prove that if u s2 see section 2 is a solution of the above equation then u c

dirichlet and neumann eigenvalue problems on domains in euclidean spaces abstract we obtain here some inequalities for the eigenvalues of dirichlet and neumann value problems for general classes of operators or system of operators acting in l 2 or l 2 c m r d d 1

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

hausdorff dimension of singular vectors we prove that the set of singular vectors in rd d 2 has hausdorff dimension d2d 1 and that the hausdorff dimension of the set of e dirichlet improvable vectors in rd is roughly d2d 1 plus a power of e between d2 and d as a corollary the set of divergent trajectories of the flow by diag et et e dt acting on sld 1 r sld 1 z has hausdorff codimension dd 1 these results extend the work of the first author

semiclassical solutions for the nonlinear schrodinger maxwell equations abstract in this paper by using variational methods and critical point theory we study the existence of semiclassical solutions for the following nonlinear schrodinger maxwell equations e 2 u v x u u f x u in r 3 4 u 2 in r 3 where e 0 and v x 0 for all x r 3 under some weaker assumptions on v and f we prove that the system has at least one nontrivial solution for sufficient small e 0

dichotomy for the hausdorff dimension of the set of nonergodic directions given an irrational 0 1 we consider billiards in the table p formed by a tfrac 1 2 times1 rectangle with a horizontal barrier of length frac 1 lambda 2 with one end touching at the midpoint of a vertical side let ne p be the set of such that the flow on p in direction is not ergodic we show that the hausdorff dimension of ne p can only take on the values 0 and tfrac 1 2 depending on the summability of the series sum k frac log log q k 1 q k where q k is the sequence of denominators of the continued fraction expansion of more specifically we prove that the hausdorff dimension is frac 1 2 if this series converges and 0 otherwise this extends earlier results of boshernitzan and cheung

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles