We study sign changing solutions to equations of the form - 2 u + u = f (u) in B, u = 0 on B, where B is the unit ball in R N (N 2), is a positive constant and f (u) behaves like |u| p-1 u (but not necessarily odd) with 1 < p < (N + 2)/(N -2) if N 3, and 1 < p < if N = 2. We show that for any given positive integer n, this problem has a sign changing radial solution v(|x|) which changes sign at exactly n spheres n j=1 {|x| = j }, where 0 < 1 < < n < 1 and as 0, j 0 and v(r) 0 uniformly on compact subsets of (0, 1]. Moreover, given any sequence k 0, there is a subsequence k i such that u(|x|) := v(|x|) converges to some U in C 1 loc (R N ) along this subsequence, and U = U (|x|) is a radial sign changing solution of

We investigate the large-time behavior of solutions of the Cauchy problem for Hamilton-Jacobi equations on the real line R.We establish a result on convergence of the solutions to asymptotic solutions as time t goes to infinity.

The purpose of this paper is to establish some basic interior estimates of Cauchy-Schauder type for the heat flow associated with a system of smooth vector fields satisfying Hormander's finite rank condition

For an m-dimensional linear process of the form Xt = j=0 A j Z t-j , where {Zt} is a sequence of stationary m-dimensional negatively associated random vectors with EZt = O and E||Zt|| 2 < , we prove the central limit theorems.

denote the kth elementary symmetric function, and let Γk denote the connected component of {λ ∈ R : σk(λ) > 0} containing the positive cone {λ ∈ R : λ1 > 0, · · · , λn > 0}. It is well known that Γk = {λ ∈ R : σl(λ) > 0, 1 ≤ l ≤ k}. Let S denote the set of n× n real symmetric matrices. For any A ∈ S we denote by λ(A) the eigenvalues of A. Throughout this note we will assume that Γ ⊂ R is an open convex symmetric cone with vertex at the origin satisfying Γn ⊂ Γ ⊂ Γ1. Moreover, we also assume that f is a continuous function defined on Γ verifying the following properties:

Motivated by questions on the preconditioning of spectral methods, and indepen- dently of the extensive literature on the approximation of zeroes of orthogonal polynomials, either by the Sturm method, or by the descent method, we develop a stationary phase-like technique for calculating asymptotics of Legendre polynomials. The difference with the classical stationary phase method is that the phase is a nonlinear function of the large parameter and the integration variable, instead of being a product of the large parameter by a function of the integration variable. We then use an implicit functions theorem for approximating the zeroes of the derivatives of Legen- dre polynomials. This result is used for proving order and consistency of the residual smoothing scheme (1), (19).

In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H. Our solutions are conservative, in the sense that the total energy ∫ (u + ux) dx remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result. 1 Introduction The Camassa-Holm equation has the form ut − utxx + 3uux = 2uxuxx + uuxxx . Equivalently, it can be written as a scalar conservation law with an additional integro-differential term: ut + (u /2)x + Px = 0 , (1.1) where P is defined as a convolution: P . = 1 2 e−|x| ∗ ( u + ux 2 ) . (1.2) For the physical motivations of this equation we refer to [CH], [CM1], [CM2], [J]. Earlier results on the existence and uniqueness of solutions can be found in [XZ1], [XZ2]. One can regard (1.1) as an evolution equation on a space of absolutely continuous functions with derivatives ux ∈ L. In the smooth case, differentiating (1.1) w.r.t. x one obtains uxt + uuxx + u 2 x − ( u + ux 2 ) + P = 0 . (1.3)

We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schr\odinger equation with an oscillatory potential. The problem therefore contains three scales, $\eps$ the oscillation length, $\theta$ the regularization parameter, $\delta$ the potential lattice. We prove that the homogenized limit (as $\eps$ vanishes) of this equation is a scattering equation with discrete jumps. As $\delta$ vanishes, the discrete scattering kernel boils down to a standard regular scattering kernel. As $\theta$ vanishes we recover the quantum scattering operator with collisions preserving energy sphere.

An exact formula for the photonic band structure (and hence the dispersion relation) is found for a certain class of periodic media. These media have the special property that the spatially fluctuating part of their complex electrical permittivity and magnetic permeability, when Fourier transformed, vanishes in half of Fourier space. As a result the band structure is proved to be exactly the same as for a lossy homogeneous medium, at least when the frequency is not too high.

We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions. The main feature of this modified particle method is that we do not require different families of characteristics to meet at one point. In the modified particle method, we update the ith component of the solution along its own characteristics, and interpolate the other components of the solution from their own characteristic points to the ith characteristic point. We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model. The same result alsoapplies to the non-resonant Broadwell model. Numerical evidence suggests that the modified particle method also converges essentially independent of the small scale for the original Broadwell model if a cubic spline interpolation is used.