sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

global existence and convergence for a higher order flow in conformal geometry an important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function e g a constant in two dimensions the uniformization theorem assures the existence of a conformal metric with constant gauss curvature moreover j moser 20 proved that for every positive function f on s2 satisfying f x f x for all x e s2 there exists a conformal metric on s2 whose gauss curvature is equal to f a natural conformal invariant in dimension four is 1 2 q ar r2 3 irici2 where r denotes the scalar curvature and ric the ricci tensor this formula can also be written in the form

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

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

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

existence of infinitely many periodic solutions for the radially symmetric wave equation with resonance abstract in this paper we consider the periodic dirichlet problem for a forced nonlinear wave equation with resonance u t t u u a t x u p 1 u in a n dimensional ball under some suitable assumptions on p and a t x we prove the existence of infinitely many radially symmetric time periodic solutions for the problem by variational methods

infinitely many periodic solutions for a semilinear wave equation in a ball in rn abstract in this paper we study the wave equation with sub linearity in a ball with periodic dirichlet conditions combining the minimax principle with an approximation argument we prove the existence of infinitely many time periodic solutions for the problem the proof also relies on the structure of the spectrum of the linearized problem

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

a property of exponential stability in nonlinear wave equations near the fundamental linear mode abstract we consider the nonlinear wave equation utt c2uxx u u t 0 u t 0 where is an analytic function satisfying 0 0 and 00 0 for each value of the harmonic energy e i e energy when 0 we consider the closed curve e which is the phase space trajectory of the linear mode with lowest frequency if the perturbation parameter and the harmonic energy e are small enough then near e we construct by means of a suitable simplified equation a closed curve e with the following property provided the distance between the initial datum and such a curve is small enough the corresponding solution remains close to e up to times exponentially long with 1 eve cannot expect the curves e to be trajectories of solutions of such curves depend smoothly on the parameters e and the above result is deduced from a general theorem on generic hamiltonian perturbations of completely resonant linear systems

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

free and forced vibrations of nonlinear wave equations in ball first we shall deal with the free vibrations of a nonlinear radially symmetric wave equation t2 u f r u in n dimensional ball ba with center at the origin and radius a where f is smooth monotone decreasing in u and satisfies f r 0 0 f r u has asymptotic properties f r u o u3 u 0andu for n 1 3 we shall show the existence of infinitely many radially symmetric time periodic solutions with different periods of irrational multiple of a second we shall deal with bvp for a forced nonlinear wave equation t2 u eg r t u where g is t periodic in t and e is a small parameter under some diophantine condition on a t we shall show the existence of time periodic solutions of the bvp for 1 n 5 we shall construct infinitely many t satisfying the above diophantine inequality using asymptotic expansions of the zero points of the bessel functions

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

comparisons of two moments based estimators that utilize historical and paleoflood data for the log pearson type iii distribution 1 the expected moments algorithm ema cohn et al 1997 and the bulletin 17b interagency committee on water data 1982 historical weighting procedure b17h for the log pearson type iii distribution are compared by monte carlo computer simulation for cases in which historical and or paleoflood data are available the relative performance of the estimators was explored for three cases fixed threshold exceedances a fixed number of large floods and floods generated from a different parent distribution ema can effectively incorporate four types of historical and paleoflood data floods where the discharge is explicitly known unknown discharges below a single threshold floods with unknown discharge that exceed some level and floods with discharges described in a range the b17h estimator can utilize only the first two types of historical information including historical paleoflood data in the simulation experiments significantly improved the quantile estimates in terms of mean square error and bias relative to using gage data alone ema performed significantly better than b17h in nearly all cases considered b17h performed as well as ema for estimating x100 in some limited fixed threshold exceedance cases ema performed comparatively much better in other fixed threshold situations for the single large flood case and in cases when estimating extreme floods equal to or greater than x500 b17h did not fully utilize historical information when the historical period exceeded 200 years robustness studies using gev simulated data confirmed that ema performed better than b17h overall ema is preferred to b17h when historical and paleoflood data are available for flood frequency analysis

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric