Second-Order Elliptic Boundary Value Problems with a Small Leading Part

Abstract

A typical second-order linear elliptic mixed (Robin) boundary value problem (boundary value problem), which arises in many modern applications, is to solve

$$\begin{aligned} \varepsilon \Delta u_\varepsilon ({\varvec{x}})+{\varvec{b}}({\varvec{x}})\cdot \nabla u_\varepsilon ({\varvec{x}})=&\, f({\varvec{x}})\text{ for }\ {\varvec{x}}\in \Omega \end{aligned}$$

$$\begin{aligned} \beta ({\varvec{x}})u_\varepsilon ({\varvec{x}})+\alpha ({\varvec{x}})\frac{\partial u_\varepsilon ({\varvec{x}})}{\partial n({\varvec{x}})} =&\, g({\varvec{x}}) \text{ for }\ {\varvec{x}}\in \partial \Omega , \end{aligned}$$

where $\Omega$ is a domain in the Euclidean space $\mathbb R^{d}$, whose boundary $\partial \Omega$ is sufficiently smooth.