Asymptotic Analysis

We develop a microlocal study for scalar classical analytic pseudo-differential operators of principal type. Next, we reduce a system of pseudo-differential operators to the normal form. Then we obtain asymptotic expansion and the exact exponential decay rate for the width of resonances arising from microlocal tunneling effect.

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the nonautonomous first-order problem

u˙+ℳu=g(t),  t∈R₊ ,

and for the second-order problem

u¨+u˙+ℳu=g(t),  t∈R₊,

where ℳ is a gradient operator and g is a function which tends to 0 when t tends to ∞.

For certain compactly supported metric and/or potential perturbations of the Laplacian on Hⁿ⁺¹, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in Hⁿ⁺¹, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.

This article is concerned with the non-existence of stable solutions for a fourth-order semilinear elliptic equation Δ²u=f(u) in R^N, where f is a smooth nonlinearity. We establish the non-existence of stable radial solutions which verify decay conditions at infinity. Our Liouville-type results do not depend on the specific nonlinearity f. Moreover, in low dimensions and under no radial symmetry assumption, we prove Liouville-type results when f is increasing.

It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1–23 that singularities of solutions for Hamilton–Jacobi equations will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized characteristics. It is natural to think that (*) generalized characteristics associated to viscosity solutions of oscillatory Hamilton–Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (*) fails without those assumptions. Some issues related to the weak KAM theory are also discussed.

We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one.