Boundary layer

Definition

Let $a < b$ and let $(y_\varepsilon)_{0 < \varepsilon < \varepsilon_0}$ be a family of functions $[a, b] \to \R^n$, for example the solutions of a singularly perturbed boundary value problem. Assume there is a continuous function $y_0 \colon (a, b] \to \R^n$, the outer limit, such that $y_\varepsilon \to y_0$ uniformly on $[a + \eta, b]$ for every $\eta \in (0, b - a)$. The family has a boundary layer at $a$ if the convergence is not uniform on $(a, b]$, or $y_\varepsilon(a)$ does not tend to $\lim_{x \to a^+} y_0(x)$. Let $\delta(\varepsilon) > 0$ with $\delta(\varepsilon) \to 0$, and consider the rescaled functions $X \mapsto y_\varepsilon(a + \delta(\varepsilon) X)$. If they converge to a nonconstant function, uniformly on compact subsets of $[0, \infty)$, then $\delta$ is a thickness of the layer. A boundary layer at $b$ is defined in the same way.