On a conjecture of Fefferman and Graham

In 1985, Fefferman and Graham have constructed a local embedding of an arbitrary real-analytic manifold, of odd dimension n, into a Ricci-flat manifold of dimension n+2 admitting a homothety. They conjectured that their result remains valid in even dimensions, if logarithms are allowed in the expansion of the metric. In this paper, we (i) prove that such expansions exist and converge and (ii) establish the degree of non-uniqueness of the solution in terms of the coefficients in the expansion.     

The relativistic Doppler broadening of the line absorption profile

The classical results of Doppler broadening of the line absorption profile are generalized to a relativistic gas in thermal equilibrium by taking into account the relativistic variance of the volume absorption coefficients of the gas, as derived by L.H. Thomas. This variance produces a small correction, even in the non-relativistic approximation.     

Boundary behavior in the Loewner-Nirenberg problem

Let Ω⊂Rⁿ be a bounded domain of class C^2+α, 0<α<1. We show that if n?3 and u_Ω is the maximal solution of equation Δu=n(n-2)u^(n+2)/(n-2) in Ω, then the hyperbolic radius is of class C^2+α up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.     

The redistribution function for scattering by relativistic electrons

The frequency redistribution function r(ν_i, ν_f) for scattering by relativistic electrons is derived, when this scattering is isotropic in the electron rest frame and recoil effects are neglected. Unlike previous derivations, we obtain for r(ν_i, ν_f) an expression which reduces exactly to its classical value for a non-relativistic gas.     

Applications of the division theorem inH s

We prove a version of the division theorem in Sobolev spaces with an estimate of the constant ass tends to infinity. We then apply it to derive spatial decay estimates for time-periodic solutions of linear wave equations in one space dimension and to prove that the space of decaying solutions is finite-dimensional. The main point is to show that some of the arguments used to analyze embedded eigenvalues of Schrödinger operators can be extended to cases where positivity arguments are not available. This has implications for nonlinear Klein-Gordon equations. A different approach, based on the proof of the stable manifold theorem, is also worked out, under slightly different assumptions.