Asymptotic Analysis

In this note the $S$ ‐matrix naturally associated with a singularly perturbed three‐dimensional system of linear differential equations without turning point on the real axis is considered. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circumstances. In particular, we show how to compute asymptotically all exponentially small off‐diagonal elements of the corresponding $S$ ‐matrix.

Location of the discrete and continuous spectrum of the operator corresponding to a boundary value problem for an elliptic system of equations in an unbounded cylinder is studied. Stability of multi‐dimensional travelling waves with respect to small perturbations is proved. These results allow us to prove global stability of travelling waves, i.e., that they describe large time asymptotic of solutions of the initial‐boundary value problem for a class of initial conditions, and to obtain a minimax representation for the wave velocity.

In this article, we study some partial differential equations with Neumann nonlinear boundary conditions, related to the problem of the best constant from $W^{1,p}(\varOmega)$ to $L^{p^\star}(\curpartial\varOmega)$ , where $p^\star$ is the critical Sobolev exponent for the embedding of $W^{1,p}(\varOmega)$ into $L^{q}(\curpartial\varOmega)$ ( $\varOmega $ is a bounded $\mathcal{C}^1$ open set and $p<N$ ).